PLEASE ANSWER BOTH QUESTIONS!! WILL RATE HIGH! I APPRECIATE YOU!! Consider the f

Question

PLEASE ANSWER BOTH QUESTIONS!! WILL RATE HIGH! I APPRECIATE YOU!!

Consider the following alternating series, S- 22k-2 (a) Estimate the sum of the series by adding the first four terms S4 (b) Use the error estimate for alternating series to estimate the error S S4 S (c) Using Theorem 8.19 on page 536, how many terms do you need to obtain three-place accuracy? N- Note for three-place accuracy you need IS SNI S .0005. Consider the following alternating series, S (a) Estimate the sum of the series by adding the first four terms S (b) Use the error estimate for alternating series to estimate the error IS - S4lS (c) Using Theorem 8.19 on page 536, how many terms do you need to obtain three-place accuracy? N- e-1 Note for three-place accuracy you need IS - SNl S .0005.

Explanation / Answer

Lets find upto 4 :
1/1 - 1/4 + 1/16 - 1/64
i.e 0.796875 ---> ANS

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b)
Error estimate :
We know that |s - s4| <= |a5|

Now, wjat's a5?
It is obtained when we put k = 5

(-1)^6 / (2^8)

i.e 1/256

So, 1/256 ----> ANS

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c)
We have |a(n+1)|
as 1 / (2^(2k)

Now, we need
1/2^(2k) <= 0.0005

So, 2^(2k) >= 2000

4^k >= 2000

k = 6

So,
n = 6 ----> ANS

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a)
-1 + 1/4 - 1/9 + 1/16

S4 = -115/144 ----> ANS

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b)
|S - s4| <= a5

And what's a5 ?
(-1)^5 / 5^2, i.e -1/25

So, 1/25 ----> ANS

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c)
1/(k+1)^2 <= 0.0005

(k+1)^2 >= 2000

k + 1 >= 44.7213595499957939

k >= 43.7213595499957939

So, 44

n = 44 ----> ANS

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