 Calculus Volume 2

# 5.4Comparison Tests

Calculus Volume 25.4 Comparison Tests

### Learning Objectives

• 5.4.1 Use the comparison test to test a series for convergence.
• 5.4.2 Use the limit comparison test to determine convergence of a series.

We have seen that the integral test allows us to determine the convergence or divergence of a series by comparing it to a related improper integral. In this section, we show how to use comparison tests to determine the convergence or divergence of a series by comparing it to a series whose convergence or divergence is known. Typically these tests are used to determine convergence of series that are similar to geometric series or p-series.

### Comparison Test

In the preceding two sections, we discussed two large classes of series: geometric series and p-series. We know exactly when these series converge and when they diverge. Here we show how to use the convergence or divergence of these series to prove convergence or divergence for other series, using a method called the comparison test.

For example, consider the series

$∑n=1∞1n2+1.∑n=1∞1n2+1.$

This series looks similar to the convergent series

$∑n=1∞1n2.∑n=1∞1n2.$

Since the terms in each of the series are positive, the sequence of partial sums for each series is monotone increasing. Furthermore, since

$0<1n2+1<1n20<1n2+1<1n2$

for all positive integers $n,n,$ the $kthkth$ partial sum $SkSk$ of $∑n=1∞1n2+1∑n=1∞1n2+1$ satisfies

$Sk=∑n=1k1n2+1<∑n=1k1n2<∑n=1∞1n2.Sk=∑n=1k1n2+1<∑n=1k1n2<∑n=1∞1n2.$

(See Figure 5.16(a) and Table 5.1.) Since the series on the right converges, the sequence ${Sk}{Sk}$ is bounded above. We conclude that ${Sk}{Sk}$ is a monotone increasing sequence that is bounded above. Therefore, by the Monotone Convergence Theorem, ${Sk}{Sk}$ converges, and thus

$∑n=1∞1n2+1∑n=1∞1n2+1$

converges.

Similarly, consider the series

$∑n=1∞1n−1/2.∑n=1∞1n−1/2.$

This series looks similar to the divergent series

$∑n=1∞1n.∑n=1∞1n.$

The sequence of partial sums for each series is monotone increasing and

$1n−1/2>1n>01n−1/2>1n>0$

for every positive integer $n.n.$ Therefore, the $kthkth$ partial sum $SkSk$ of $∑n=1∞1n−1/2∑n=1∞1n−1/2$ satisfies

$Sk=∑n=1k1n−1/2>∑n=1k1n.Sk=∑n=1k1n−1/2>∑n=1k1n.$

(See Figure 5.16(b) and Table 5.2.) Since the series $∑n=1∞1/n∑n=1∞1/n$ diverges to infinity, the sequence of partial sums $∑n=1k1/n∑n=1k1/n$ is unbounded. Consequently, ${Sk}{Sk}$ is an unbounded sequence, and therefore diverges. We conclude that

$∑n=1∞1n−1/2∑n=1∞1n−1/2$

diverges.

Figure 5.16 (a) Each of the partial sums for the given series is less than the corresponding partial sum for the converging $p − series . p − series .$ (b) Each of the partial sums for the given series is greater than the corresponding partial sum for the diverging harmonic series.
$kk$ $11$ $22$ $33$ $44$ $55$ $66$ $77$ $88$
$∑n=1k1n2+1∑n=1k1n2+1$ $0.50.5$ $0.70.7$ $0.80.8$ $0.85880.8588$ $0.89730.8973$ $0.92430.9243$ $0.94430.9443$ $0.95970.9597$
$∑n=1k1n2∑n=1k1n2$ $11$ $1.251.25$ $1.36111.3611$ $1.42361.4236$ $1.46361.4636$ $1.49141.4914$ $1.51181.5118$ $1.52741.5274$
Table 5.1 Comparing a series with a p-series (p = 2)
$kk$ $11$ $22$ $33$ $44$ $55$ $66$ $77$ $88$
$∑n=1k1n−1/2∑n=1k1n−1/2$ $22$ $2.66672.6667$ $3.06673.0667$ $3.35243.3524$ $3.57463.5746$ $3.75643.7564$ $3.91033.9103$ $4.04364.0436$
$∑n=1k1n∑n=1k1n$ $11$ $1.51.5$ $1.83331.8333$ $2.09332.0933$ $2.28332.2833$ $2.452.45$ $2.59292.5929$ $2.71792.7179$
Table 5.2 Comparing a series with the harmonic series

### Theorem5.11

#### Comparison Test

1. Suppose there exists an integer $NN$ such that $0≤an≤bn0≤an≤bn$ for all $n≥N.n≥N.$ If $∑n=1∞bn∑n=1∞bn$ converges, then $∑n=1∞an∑n=1∞an$ converges.
2. Suppose there exists an integer $NN$ such that $an≥bn≥0an≥bn≥0$ for all $n≥N.n≥N.$ If $∑n=1∞bn∑n=1∞bn$ diverges, then $∑n=1∞an∑n=1∞an$ diverges.

#### Proof

We prove part i. The proof of part ii. is the contrapositive of part i. Let ${Sk}{Sk}$ be the sequence of partial sums associated with $∑n=1∞an,∑n=1∞an,$ and let $L=∑n=1∞bn.L=∑n=1∞bn.$ Since the terms $an≥0,an≥0,$

$Sk=a1+a2+⋯+ak≤a1+a2+⋯+ak+ak+1=Sk+1.Sk=a1+a2+⋯+ak≤a1+a2+⋯+ak+ak+1=Sk+1.$

Therefore, the sequence of partial sums is increasing. Further, since $an≤bnan≤bn$ for all $n≥N,n≥N,$ then

$∑n=Nkan≤∑n=Nkbn≤∑n=1∞bn=L.∑n=Nkan≤∑n=Nkbn≤∑n=1∞bn=L.$

Therefore, for all $k≥1,k≥1,$

$Sk=(a1+a2+⋯+aN−1)+∑n=Nkan≤(a1+a2+⋯+aN−1)+L.Sk=(a1+a2+⋯+aN−1)+∑n=Nkan≤(a1+a2+⋯+aN−1)+L.$

Since $a1+a2+⋯+aN−1a1+a2+⋯+aN−1$ is a finite number, we conclude that the sequence ${Sk}{Sk}$ is bounded above. Therefore, ${Sk}{Sk}$ is an increasing sequence that is bounded above. By the Monotone Convergence Theorem, we conclude that ${Sk}{Sk}$ converges, and therefore the series $∑n=1∞an∑n=1∞an$ converges.

To use the comparison test to determine the convergence or divergence of a series $∑n=1∞an,∑n=1∞an,$ it is necessary to find a suitable series with which to compare it. Since we know the convergence properties of geometric series and p-series, these series are often used. If there exists an integer $NN$ such that for all $n≥N,n≥N,$ each term $anan$ is less than each corresponding term of a known convergent series, then $∑n=1∞an∑n=1∞an$ converges. Similarly, if there exists an integer $NN$ such that for all $n≥N,n≥N,$ each term $anan$ is greater than each corresponding term of a known divergent series, then $∑n=1∞an∑n=1∞an$ diverges.

### Example 5.17

#### Using the Comparison Test

For each of the following series, use the comparison test to determine whether the series converges or diverges.

1. $∑n=1∞1n3+3n+1∑n=1∞1n3+3n+1$
2. $∑n=1∞12n+1∑n=1∞12n+1$
3. $∑n=2∞1ln(n)∑n=2∞1ln(n)$

### Checkpoint5.16

Use the comparison test to determine if the series $∑n=1∞nn3+n+1∑n=1∞nn3+n+1$ converges or diverges.

### Limit Comparison Test

The comparison test works nicely if we can find a comparable series satisfying the hypothesis of the test. However, sometimes finding an appropriate series can be difficult. Consider the series

$∑n=2∞1n2−1.∑n=2∞1n2−1.$

It is natural to compare this series with the convergent series

$∑n=2∞1n2.∑n=2∞1n2.$

However, this series does not satisfy the hypothesis necessary to use the comparison test because

$1n2−1>1n21n2−1>1n2$

for all integers $n≥2.n≥2.$ Although we could look for a different series with which to compare $∑n=2∞1/(n2−1),∑n=2∞1/(n2−1),$ instead we show how we can use the limit comparison test to compare

$∑n=2∞1n2−1and∑n=2∞1n2.∑n=2∞1n2−1and∑n=2∞1n2.$

Let us examine the idea behind the limit comparison test. Consider two series $∑n=1∞an∑n=1∞an$ and $∑n=1∞bn.∑n=1∞bn.$ with positive terms $anandbnanandbn$ and evaluate

$limn→∞anbn.limn→∞anbn.$

If

$limn→∞anbn=L≠0,limn→∞anbn=L≠0,$

then, for $nn$ sufficiently large, $an≈Lbn.an≈Lbn.$ Therefore, either both series converge or both series diverge. For the series $∑n=2∞1/(n2−1)∑n=2∞1/(n2−1)$ and $∑n=2∞1/n2,∑n=2∞1/n2,$ we see that

$limn→∞1/(n2−1)1/n2=limn→∞n2n2−1=1.limn→∞1/(n2−1)1/n2=limn→∞n2n2−1=1.$

Since $∑n=2∞1/n2∑n=2∞1/n2$ converges, we conclude that

$∑n=2∞1n2−1∑n=2∞1n2−1$

converges.

The limit comparison test can be used in two other cases. Suppose

$limn→∞anbn=0.limn→∞anbn=0.$

In this case, ${an/bn}{an/bn}$ is a bounded sequence. As a result, there exists a constant $MM$ such that $an≤Mbn.an≤Mbn.$ Therefore, if $∑n=1∞bn∑n=1∞bn$ converges, then $∑n=1∞an∑n=1∞an$ converges. On the other hand, suppose

$limn→∞anbn=∞.limn→∞anbn=∞.$

In this case, ${an/bn}{an/bn}$ is an unbounded sequence. Therefore, for every constant $MM$ there exists an integer $NN$ such that $an≥Mbnan≥Mbn$ for all $n≥N.n≥N.$ Therefore, if $∑n=1∞bn∑n=1∞bn$ diverges, then $∑n=1∞an∑n=1∞an$ diverges as well.

### Theorem5.12

#### Limit Comparison Test

Let $an,bn≥0an,bn≥0$ for all $n≥1.n≥1.$

1. If $limn→∞an/bn=L≠0,limn→∞an/bn=L≠0,$ then $∑n=1∞an∑n=1∞an$ and $∑n=1∞bn∑n=1∞bn$ both converge or both diverge.
2. If $limn→∞an/bn=0limn→∞an/bn=0$ and $∑n=1∞bn∑n=1∞bn$ converges, then $∑n=1∞an∑n=1∞an$ converges.
3. If $limn→∞an/bn=∞limn→∞an/bn=∞$ and $∑n=1∞bn∑n=1∞bn$ diverges, then $∑n=1∞an∑n=1∞an$ diverges.

Note that if $an/bn→0an/bn→0$ and $∑n=1∞bn∑n=1∞bn$ diverges, the limit comparison test gives no information. Similarly, if $an/bn→∞an/bn→∞$ and $∑n=1∞bn∑n=1∞bn$ converges, the test also provides no information. For example, consider the two series $∑n=1∞1/n∑n=1∞1/n$ and $∑n=1∞1/n2.∑n=1∞1/n2.$ These series are both p-series with $p=1/2p=1/2$ and $p=2,p=2,$ respectively. Since $p=1/2<1,p=1/2<1,$ the series $∑n=1∞1/n∑n=1∞1/n$ diverges. On the other hand, since $p=2>1,p=2>1,$ the series $∑n=1∞1/n2∑n=1∞1/n2$ converges. However, suppose we attempted to apply the limit comparison test, using the convergent $p−seriesp−series$ $∑n=1∞1/n3∑n=1∞1/n3$ as our comparison series. First, we see that

$1/n1/n3=n3n=n5/2→∞asn→∞.1/n1/n3=n3n=n5/2→∞asn→∞.$

Similarly, we see that

$1/n21/n3=n→∞asn→∞.1/n21/n3=n→∞asn→∞.$

Therefore, if $an/bn→∞an/bn→∞$ when $∑n=1∞bn∑n=1∞bn$ converges, we do not gain any information on the convergence or divergence of $∑n=1∞an.∑n=1∞an.$

### Example 5.18

#### Using the Limit Comparison Test

For each of the following series, use the limit comparison test to determine whether the series converges or diverges. If the test does not apply, say so.

1. $∑n=1∞1n+1∑n=1∞1n+1$
2. $∑n=1∞2n+13n∑n=1∞2n+13n$
3. $∑n=1∞ln(n)n2∑n=1∞ln(n)n2$

### Checkpoint5.17

Use the limit comparison test to determine whether the series $∑n=1∞5n3n+2∑n=1∞5n3n+2$ converges or diverges.

### Section 5.4 Exercises

Use the comparison test to determine whether the following series converge.

194 .

$∑n=1∞an∑n=1∞an$ where $an=2n(n+1)an=2n(n+1)$

195 .

$∑n=1∞an∑n=1∞an$ where $an=1n(n+1/2)an=1n(n+1/2)$

196 .

$∑ n = 1 ∞ 1 2 ( n + 1 ) ∑ n = 1 ∞ 1 2 ( n + 1 )$

197 .

$∑ n = 1 ∞ 1 2 n − 1 ∑ n = 1 ∞ 1 2 n − 1$

198 .

$∑ n = 2 ∞ 1 ( n ln n ) 2 ∑ n = 2 ∞ 1 ( n ln n ) 2$

199 .

$∑ n = 1 ∞ n ! ( n + 2 ) ! ∑ n = 1 ∞ n ! ( n + 2 ) !$

200 .

$∑ n = 1 ∞ 1 n ! ∑ n = 1 ∞ 1 n !$

201 .

$∑ n = 1 ∞ sin ( 1 / n ) n ∑ n = 1 ∞ sin ( 1 / n ) n$

202 .

$∑ n = 1 ∞ sin 2 n n 2 ∑ n = 1 ∞ sin 2 n n 2$

203 .

$∑ n = 1 ∞ sin ( 1 / n ) n ∑ n = 1 ∞ sin ( 1 / n ) n$

204 .

$∑ n = 1 ∞ n 1.2 − 1 n 2.3 + 1 ∑ n = 1 ∞ n 1.2 − 1 n 2.3 + 1$

205 .

$∑ n = 1 ∞ n + 1 − n n ∑ n = 1 ∞ n + 1 − n n$

206 .

$∑ n = 1 ∞ n 4 n 4 + n 2 3 ∑ n = 1 ∞ n 4 n 4 + n 2 3$

Use the limit comparison test to determine whether each of the following series converges or diverges.

207 .

$∑ n = 1 ∞ ( ln n n ) 2 ∑ n = 1 ∞ ( ln n n ) 2$

208 .

$∑ n = 1 ∞ ( ln n n 0.6 ) 2 ∑ n = 1 ∞ ( ln n n 0.6 ) 2$

209 .

$∑ n = 1 ∞ ln ( 1 + 1 n ) n ∑ n = 1 ∞ ln ( 1 + 1 n ) n$

210 .

$∑ n = 1 ∞ ln ( 1 + 1 n 2 ) ∑ n = 1 ∞ ln ( 1 + 1 n 2 )$

211 .

$∑ n = 1 ∞ 1 4 n − 3 n ∑ n = 1 ∞ 1 4 n − 3 n$

212 .

$∑ n = 1 ∞ 1 n 2 − n sin n ∑ n = 1 ∞ 1 n 2 − n sin n$

213 .

$∑ n = 1 ∞ 1 e ( 1.1 ) n − 3 n ∑ n = 1 ∞ 1 e ( 1.1 ) n − 3 n$

214 .

$∑ n = 1 ∞ 1 e ( 1.01 ) n − 3 n ∑ n = 1 ∞ 1 e ( 1.01 ) n − 3 n$

215 .

$∑ n = 1 ∞ 1 n 1 + 1 / n ∑ n = 1 ∞ 1 n 1 + 1 / n$

216 .

$∑ n = 1 ∞ 1 2 1 + 1 / n n 1 + 1 / n ∑ n = 1 ∞ 1 2 1 + 1 / n n 1 + 1 / n$

217 .

$∑ n = 1 ∞ ( 1 n − sin ( 1 n ) ) ∑ n = 1 ∞ ( 1 n − sin ( 1 n ) )$

218 .

$∑ n = 1 ∞ ( 1 − cos ( 1 n ) ) ∑ n = 1 ∞ ( 1 − cos ( 1 n ) )$

219 .

$∑ n = 1 ∞ 1 n ( π 2 − tan −1 n ) ∑ n = 1 ∞ 1 n ( π 2 − tan −1 n )$

220 .

$∑n=1∞(1−1n)n.n∑n=1∞(1−1n)n.n$ (Hint:$(1−1n)n→1/e.)(1−1n)n→1/e.)$

221 .

$∑n=1∞(1−e−1/n)∑n=1∞(1−e−1/n)$ (Hint:$1/e≈(1−1/n)n,1/e≈(1−1/n)n,$ so $1−e−1/n≈1/n.)1−e−1/n≈1/n.)$

222 .

Does $∑n=2∞1(lnn)p∑n=2∞1(lnn)p$ converge if $pp$ is large enough? If so, for which $p?p?$

223 .

Does $∑n=1∞((lnn)n)p∑n=1∞((lnn)n)p$ converge if $pp$ is large enough? If so, for which $p?p?$

224 .

For which $pp$ does the series $∑n=1∞2pn/3n∑n=1∞2pn/3n$ converge?

225 .

For which $p>0p>0$ does the series $∑n=1∞np2n∑n=1∞np2n$ converge?

226 .

For which $r>0r>0$ does the series $∑n=1∞rn22n∑n=1∞rn22n$ converge?

227 .

For which $r>0r>0$ does the series $∑n=1∞2nrn2∑n=1∞2nrn2$ converge?

228 .

Find all values of $pp$ and $qq$ such that $∑n=1∞np(n!)q∑n=1∞np(n!)q$ converges.

229 .

Does $∑n=1∞sin2(nr/2)n∑n=1∞sin2(nr/2)n$ converge or diverge? Explain.

230 .

Explain why, for each $n,n,$ at least one of ${|sinn|,|sin(n+1)|,...,|sinn+6|}{|sinn|,|sin(n+1)|,...,|sinn+6|}$ is larger than $1/2.1/2.$ Use this relation to test convergence of $∑n=1∞|sinn|n.∑n=1∞|sinn|n.$

231 .

Suppose that $an≥0an≥0$ and $bn≥0bn≥0$ and that $∑n=1∞a2n∑n=1∞a2n$ and $∑n=1∞b2n∑n=1∞b2n$ converge. Prove that $∑n=1∞anbn∑n=1∞anbn$ converges and $∑n=1∞anbn≤12(∑n=1∞an2+∑n=1∞bn2).∑n=1∞anbn≤12(∑n=1∞an2+∑n=1∞bn2).$

232 .

Does $∑n=1∞2−lnlnn∑n=1∞2−lnlnn$ converge? (Hint: Write $2lnlnn2lnlnn$ as a power of $lnn.)lnn.)$

233 .

Does $∑n=1∞(lnn)−lnn∑n=1∞(lnn)−lnn$ converge? (Hint: Use $n=eln(n)n=eln(n)$ to compare to a $p−series.)p−series.)$

234 .

Does $∑n=2∞(lnn)−lnlnn∑n=2∞(lnn)−lnlnn$ converge? (Hint: Compare $anan$ to $1/n.)1/n.)$

235 .

Show that if $an≥0an≥0$ and $∑n=1∞an∑n=1∞an$ converges, then $∑n=1∞a2n∑n=1∞a2n$ converges. If $∑n=1∞a2n∑n=1∞a2n$ converges, does $∑n=1∞an∑n=1∞an$ necessarily converge?

236 .

Suppose that $an>0an>0$ for all $nn$ and that $∑n=1∞an∑n=1∞an$ converges. Suppose that $bnbn$ is an arbitrary sequence of zeros and ones. Does $∑n=1∞anbn∑n=1∞anbn$ necessarily converge?

237 .

Suppose that $an>0an>0$ for all $nn$ and that $∑n=1∞an∑n=1∞an$ diverges. Suppose that $bnbn$ is an arbitrary sequence of zeros and ones with infinitely many terms equal to one. Does $∑n=1∞anbn∑n=1∞anbn$ necessarily diverge?

238 .

Complete the details of the following argument: If $∑n=1∞1n∑n=1∞1n$ converges to a finite sum $s,s,$ then $12s=12+14+16+⋯12s=12+14+16+⋯$ and $s−12s=1+13+15+⋯.s−12s=1+13+15+⋯.$ Why does this lead to a contradiction?

239 .

Show that if $an≥0an≥0$ and $∑n=1∞a2n∑n=1∞a2n$ converges, then $∑n=1∞sin2(an)∑n=1∞sin2(an)$ converges.

240 .

Suppose that $an/bn→0an/bn→0$ in the comparison test, where $an≥0an≥0$ and $bn≥0.bn≥0.$ Prove that if $∑bn∑bn$ converges, then $∑an∑an$ converges.

241 .

Let $bnbn$ be an infinite sequence of zeros and ones. What is the largest possible value of $x=∑n=1∞bn/2n?x=∑n=1∞bn/2n?$

242 .

Let $dndn$ be an infinite sequence of digits, meaning $dndn$ takes values in ${0,1,…,9}.{0,1,…,9}.$ What is the largest possible value of $x=∑n=1∞dn/10nx=∑n=1∞dn/10n$ that converges?

243 .

Explain why, if $x>1/2,x>1/2,$ then $xx$ cannot be written $x=∑n=2∞bn2n(bn=0or1,b1=0).x=∑n=2∞bn2n(bn=0or1,b1=0).$

244 .

[T] Evelyn has a perfect balancing scale, an unlimited number of $1-kg1-kg$ weights, and one each of $1/2-kg,1/4-kg,1/8-kg,1/2-kg,1/4-kg,1/8-kg,$ and so on weights. She wishes to weigh a meteorite of unspecified origin to arbitrary precision. Assuming the scale is big enough, can she do it? What does this have to do with infinite series?

245 .

[T] Robert wants to know his body mass to arbitrary precision. He has a big balancing scale that works perfectly, an unlimited collection of $1-kg1-kg$ weights, and nine each of $0.1-kg,0.1-kg,$ $0.01-kg,0.001-kg,0.01-kg,0.001-kg,$ and so on weights. Assuming the scale is big enough, can he do this? What does this have to do with infinite series?

246 .

The series $∑n=1∞12n∑n=1∞12n$ is half the harmonic series and hence diverges. It is obtained from the harmonic series by deleting all terms in which $nn$ is odd. Let $m>1m>1$ be fixed. Show, more generally, that deleting all terms $1/n1/n$ where $n=mkn=mk$ for some integer $kk$ also results in a divergent series.

247 .

In view of the previous exercise, it may be surprising that a subseries of the harmonic series in which about one in every five terms is deleted might converge. A depleted harmonic series is a series obtained from $∑n=1∞1n∑n=1∞1n$ by removing any term $1/n1/n$ if a given digit, say $9,9,$ appears in the decimal expansion of $n.n.$ Argue that this depleted harmonic series converges by answering the following questions.

1. How many whole numbers $nn$ have $dd$ digits?
2. How many $d-digitd-digit$ whole numbers $h(d).h(d).$ do not contain $99$ as one or more of their digits?
3. What is the smallest $d-digitd-digit$ number $m(d)?m(d)?$
4. Explain why the deleted harmonic series is bounded by $∑d=1∞h(d)m(d).∑d=1∞h(d)m(d).$
5. Show that $∑d=1∞h(d)m(d)∑d=1∞h(d)m(d)$ converges.
248 .

Suppose that a sequence of numbers $an>0an>0$ has the property that $a1=1a1=1$ and $an+1=1n+1Sn,an+1=1n+1Sn,$ where $Sn=a1+⋯+an.Sn=a1+⋯+an.$ Can you determine whether $∑n=1∞an∑n=1∞an$ converges? (Hint: $SnSn$ is monotone.)

249 .

Suppose that a sequence of numbers $an>0an>0$ has the property that $a1=1a1=1$ and $an+1=1(n+1)2Sn,an+1=1(n+1)2Sn,$ where $Sn=a1+⋯+an.Sn=a1+⋯+an.$ Can you determine whether $∑n=1∞an∑n=1∞an$ converges? (Hint: $S2=a2+a1=a2+S1=a2+1=1+1/4=(1+1/4)S1,S2=a2+a1=a2+S1=a2+1=1+1/4=(1+1/4)S1,$ $S3=132S2+S2=(1+1/9)S2=(1+1/9)(1+1/4)S1,S3=132S2+S2=(1+1/9)S2=(1+1/9)(1+1/4)S1,$ etc. Look at $ln(Sn),ln(Sn),$ and use $ln(1+t)≤t,ln(1+t)≤t,$ $t>0.)t>0.)$

Order a print copy

As an Amazon Associate we earn from qualifying purchases.