Calculus Volume 2

# Key Concepts

Calculus Volume 2Key Concepts

### 6.1Power Series and Functions

• For a power series centered at $x=a,x=a,$ one of the following three properties hold:
1. The power series converges only at $x=a.x=a.$ In this case, we say that the radius of convergence is $R=0.R=0.$
2. The power series converges for all real numbers x. In this case, we say that the radius of convergence is $R=∞.R=∞.$
3. There is a real number R such that the series converges for $|x−a| and diverges for $|x−a|>R.|x−a|>R.$ In this case, the radius of convergence is R.
• If a power series converges on a finite interval, the series may or may not converge at the endpoints.
• The ratio test may often be used to determine the radius of convergence.
• The geometric series $∑n=0∞xn=11−x∑n=0∞xn=11−x$ for $|x|<1|x|<1$ allows us to represent certain functions using geometric series.

### 6.2Properties of Power Series

• Given two power series $∑n=0∞cnxn∑n=0∞cnxn$ and $∑n=0∞dnxn∑n=0∞dnxn$ that converge to functions f and g on a common interval I, the sum and difference of the two series converge to $f±g,f±g,$ respectively, on I. In addition, for any real number b and integer $m≥0,m≥0,$ the series $∑n=0∞bxmcnxn∑n=0∞bxmcnxn$ converges to $bxmf(x)bxmf(x)$ and the series $∑n=0∞cn(bxm)n∑n=0∞cn(bxm)n$ converges to $f(bxm)f(bxm)$ whenever bxm is in the interval I.
• Given two power series that converge on an interval $(−R,R),(−R,R),$ the Cauchy product of the two power series converges on the interval $(−R,R).(−R,R).$
• Given a power series that converges to a function f on an interval $(−R,R),(−R,R),$ the series can be differentiated term-by-term and the resulting series converges to $f′f′$ on $(−R,R).(−R,R).$ The series can also be integrated term-by-term and the resulting series converges to $∫f(x)dx∫f(x)dx$ on $(−R,R).(−R,R).$

### 6.3Taylor and Maclaurin Series

• Taylor polynomials are used to approximate functions near a value $x=a.x=a.$ Maclaurin polynomials are Taylor polynomials at $x=0.x=0.$
• The nth degree Taylor polynomials for a function $ff$ are the partial sums of the Taylor series for $f.f.$
• If a function $ff$ has a power series representation at $x=a,x=a,$ then it is given by its Taylor series at $x=a.x=a.$
• A Taylor series for $ff$ converges to $ff$ if and only if $limn→∞Rn(x)=0limn→∞Rn(x)=0$ where $Rn(x)=f(x)−pn(x).Rn(x)=f(x)−pn(x).$
• The Taylor series for ex, $sinx,sinx,$ and $cosxcosx$ converge to the respective functions for all real x.

### 6.4Working with Taylor Series

• The binomial series is the Maclaurin series for $f(x)=(1+x)r.f(x)=(1+x)r.$ It converges for $|x|<1.|x|<1.$
• Taylor series for functions can often be derived by algebraic operations with a known Taylor series or by differentiating or integrating a known Taylor series.
• Power series can be used to solve differential equations.
• Taylor series can be used to help approximate integrals that cannot be evaluated by other means.