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Calculus Volume 2

Review Exercises

Calculus Volume 2Review Exercises

Review Exercises

True or False? Justify your answer with a proof or a counterexample.

379.

If limnan=0,limnan=0, then n=1ann=1an converges.

380.

If limnan0,limnan0, then n=1ann=1an diverges.

381.

If n=1|an|n=1|an| converges, then n=1ann=1an converges.

382.

If n=12nann=12nan converges, then n=1(−2)nann=1(−2)nan converges.

Is the sequence bounded, monotone, and convergent or divergent? If it is convergent, find the limit.

383.

a n = 3 + n 2 1 n a n = 3 + n 2 1 n

384.

a n = ln ( 1 n ) a n = ln ( 1 n )

385.

a n = ln ( n + 1 ) n + 1 a n = ln ( n + 1 ) n + 1

386.

a n = 2 n + 1 5 n a n = 2 n + 1 5 n

387.

a n = ln ( cos n ) n a n = ln ( cos n ) n

Is the series convergent or divergent?

388.

n = 1 1 n 2 + 5 n + 4 n = 1 1 n 2 + 5 n + 4

389.

n = 1 ln ( n + 1 n ) n = 1 ln ( n + 1 n )

390.

n = 1 2 n n 4 n = 1 2 n n 4

391.

n = 1 e n n ! n = 1 e n n !

392.

n = 1 n ( n + 1 / n ) n = 1 n ( n + 1 / n )

Is the series convergent or divergent? If convergent, is it absolutely convergent?

393.

n = 1 ( −1 ) n n n = 1 ( −1 ) n n

394.

n = 1 ( −1 ) n n ! 3 n n = 1 ( −1 ) n n ! 3 n

395.

n = 1 ( −1 ) n n ! n n n = 1 ( −1 ) n n ! n n

396.

n = 1 sin ( n π 2 ) n = 1 sin ( n π 2 )

397.

n = 1 cos ( π n ) e n n = 1 cos ( π n ) e n

Evaluate

398.

n = 1 2 n + 4 7 n n = 1 2 n + 4 7 n

399.

n = 1 1 ( n + 1 ) ( n + 2 ) n = 1 1 ( n + 1 ) ( n + 2 )

400.

A legend from India tells that a mathematician invented chess for a king. The king enjoyed the game so much he allowed the mathematician to demand any payment. The mathematician asked for one grain of rice for the first square on the chessboard, two grains of rice for the second square on the chessboard, four grains of rice for the third square on the chessboard, and so on. Find an exact expression for the total payment (in grains of rice) requested by the mathematician. Assuming there are 30,00030,000 grains of rice in 11 pound, and 20002000 pounds in 11 ton, how many tons of rice did the mathematician attempt to receive?

The following problems consider a simple population model of the housefly, which can be exhibited by the recursive formula xn+1=bxn,xn+1=bxn, where xnxn is the population of houseflies at generation n,n, and bb is the average number of offspring per housefly who survive to the next generation. Assume a starting population x0.x0.

401.

Find limnxnlimnxn if b>1,b>1, b<1,b<1, and b=1.b=1.

402.

Find an expression for Sn=i=0nxiSn=i=0nxi in terms of bb and x0.x0. What does it physically represent?

403.

If b=34b=34 and x0=100,x0=100, find S10S10 and limnSnlimnSn

404.

For what values of bb will the series converge and diverge? What does the series converge to?

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