For each of the four series below do the following: 1. Explain why each one conv

Question

For each of the four series below do the following: 1. Explain why each one converges (use an appropriate test). 2. Using a calculator or computer, add together enough terms so that you are convinced 3. (Optional, and not so easy) Prove that your approximation really is correct to four 4. Identify the four numbers you have obtained Here are the four series you know the sum as a decimal number correct to at least four decimal places. decimal places. (They are all more-or-less famous numbers). 1 1 + --+-.. 2 3 n! 2 6 24 n-0 722 ra 1 23 46

Explanation / Answer

1)
an = (-1)^n / (2n + 1)

This is an alternating series

So, let bn = 1/(2n + 1)
which both goes to 0 as n ---> inf
and also decreases as n increases

So, by alternating series test
an CONVERGES

2)
|an+1| = 1/(2n + 1)

And we need
1/(2n + 1) < 0.00005

2n + 1 < 1/0.00005

n <= 9999.5

So, n = 9999

We have to dd upto 9999 terms
Lets see what we get upon doin' that
0.7854 ---> ANS

3)
Lets find the actual sum using a calculator
It comes out to pi/4
which is 0.7853981633974483 which upto 4 decimal places is indeed
0.7854

4)
This is pi/4

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