Proof of absolute convergent (Calculus Early Trans. 11.6 Q34)?

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Let 􏰾 an be a series with positive terms and let rn 􏰩 an􏰁1 􏰴an. Suppose that limnl􏰈 rn 􏰩 L 􏰋 1, so 􏰾an converges by the Ratio Test. As usual, we let Rn be the remainder after n terms,

thatis, Rn 􏰩an􏰁1 􏰁an􏰁2 􏰁an􏰁3 􏰁􏰇􏰇􏰇

(a) If 􏱁rn 􏱃 is a decreasing sequence and rn􏰁1 􏰋 1, show, by summing a geometric series, that

Rn 􏰆 an􏰁1 1 􏰀 rn􏰁1

(b) If 􏱁rn 􏱃 is an increasing sequence, show that

a n􏰁1

􏰭􏰮

n􏰩2 n􏰁1 􏰈n

1 n2 1􏰁 24.

􏱆 n􏰩2 􏰰ln n􏰱

n

35.

36.

37.

38.

39.

40.

n􏰩1 1􏰴n2 . Use Exer- cise 34 to estimate the error in using s5 as an approximation

n

the series

􏱆􏰈 n n􏰩1 2n

5 5 􏱏 8 5 􏱏 8 􏱏 11 􏱆􏰈 2􏱏4􏱏6􏱏􏰇􏰇􏰇􏱏􏰰2n􏰱

5 􏱏 8 􏱏 11 􏱏 14

27. 28.

30.

32.

34.

n􏰩1 􏱆􏰈

n􏰩1

n!

Use Exercise 34 to estimate the error. Prove the Root Test. [Hint for part (i): Take any number r such

that L 􏰋 r 􏰋 1 and use the fact that there is an integer N such

thatsn 􏰯an􏰯􏰋rwhenevern􏰄N.] Around 1910, the Indian mathematician Srinivasa

Ramanujan discovered the formula

1 􏰩 2s2 􏱆􏰈 􏰰4n􏰱!􏰰1103 􏰁 26390n􏰱 9801 n􏰩0 􏰰n!􏰱4 3964n

William Gosper used this series in 1985 to compute the first 17 million digits of . (a) Verify that the series is convergent. (b) How many correct decimal places of do you get if you

use just the first term of the series? What if you use two terms?

Given any series 􏰾 an, we define a series 􏰾 a􏰁n whose terms are all the positive terms of 􏰾 an and a series 􏰾 a􏰀n whose terms are all the negative terms of 􏰾 an. To be specific, we let

a􏰁n 􏰩 an 􏰁􏰯an􏰯 a􏰀n 􏰩 an 􏰀􏰯an􏰯 22

Noticethatifan 􏰉0,thena􏰁n 􏰩an anda􏰀n 􏰩0,whereasif an 􏰋0,thena􏰀n 􏰩an anda􏰁n 􏰩0. (a) If 􏰾 an is absolutely convergent, show that both of the series

􏰾 a􏰁n and 􏰾 a􏰀n are convergent. (b) If 􏰾 an is conditionally convergent, show that both of the

series 􏰾 a􏰁n and 􏰾 a􏰀n are divergent.

Prove that if 􏰾 an is a conditionally convergent series and r is any real number, then there is a rearrangement of 􏰾 an whose sum is r. [Hints: Use the notation of Exercise 39. Take just enough positive terms a􏰁n so that their sum is greater than r. Then add just enough negative terms a􏰀n so that the cumulative sum is less than r. Continue in this manner and use Theorem 11.2.6.]

1􏱏3 1􏱏3􏱏5 1􏱏3􏱏5􏱏7 1􏰀3!􏰁 5! 􏰀 7! 􏰁􏰇􏰇􏰇

n􏰀1 1􏱏3􏱏5􏱏 􏰇􏰇􏰇 􏱏􏰰2n􏰀1􏰱 􏰁􏰰􏰀1􏰱 􏰰2n 􏰀 1􏰱! 􏰁 􏰇􏰇􏰇

to the sum of the series. (b) Find a value of n so that sn is within 0.00005 of the sum.

Use this value of n to approximate the sum of the series. Use the sum of the first 10 terms to approximate the sum of

26. 2 􏰁 2􏱏6 􏰁 2􏱏6􏱏10 􏰁 2􏱏6􏱏10􏱏14 􏰁􏰇􏰇􏰇

􏱆

Rn 􏰆 1 􏰀 L

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