Co 1. Consider the Alternating Harmonic Series: 2 34 rn (a) Prove that the Alter

Question

Co 1. Consider the Alternating Harmonic Series: 2 34 rn (a) Prove that the Alternating Harmonic Series converges to a limit L (b) Find an n so that the nth partial sum of the series is within 0.01 of the limit L. (c) It turns out that L in this case is In(2). Using technology, sum the first n terms of the Alternating Harmonic Series with your value of n from part (b) and compare the result to In(2) 2. Consider the series >(-1)"+'an where an 1 if n is odd and an - 1 if n is evern (-1)"+a 111 122 3425 62 TL (a) Show that this series diverges. (b) Why does the Alternating Series Test not apply?

Explanation / Answer

We have (-1)^(n+1) / n

So let us use bn = 1/n

We know that bn is both decreasing and is approaching ZERO
as n ---> inf

So, an = (-1)^(n+1) * bn satisfies alternating series test
and is thus

CONVERGENT

And since we have proved that the series an is convergent,
it means (sigma) an must be finite

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b)
|a(n+1)| = 1/(n+1)

And acc to alternating series test, we need
1/(n+1) <= 0.01

n+1 >= 100

n >= 99

So, n = 99 terms

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c)
We find the sigma from n = 1 to 99 to be :
0.6981721793

And actual answer is ln(2)

So, error is :
|ln(2) - 0.6981721793|
i.e 0.0050249987400547

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