Question: "Determine whether three series converges or diverges. if..."

Braxton Pugh

Answered question

2021-02-21

Determine whether three series converges or diverges. if it converges, find its sum

\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{4^{n}}

brawnyN

Skilled2021-02-22Added 91 answers

1)Here the series is given by

\sum_{n = 1}^{\infty} {(- 1)}^{n} a_{n} = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{4^{n}}

.This is a alternating series with

a_{n} = \frac{1}{4^{n}} . C \leq a r l y a_{n}

is a decreasing sequence and

lim_{n \to \infty} a_{n} = lim_{n \to \infty} \frac{1}{4^{n}} = 0

Therefore by Leibnit'z rule the given series

\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{4^{n}}

is convergent.
2)Consider the series

\sum_{n = 1}^{\infty} \frac{1}{n}

.The n-th partial sum of the series is given by

S_{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}

.
Now

S_{2^{n}} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \dots + \frac{1}{2^{n}}

= 1 + \frac{1}{2} + (\frac{1}{3} + \frac{1}{4}) + (\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}) + \dots + \frac{1}{2^{n}}

\Rightarrow 1 + \frac{1}{2} + (\frac{1}{4} + \frac{1}{4}) + (\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}) + \dots + (\frac{1}{n} + \dots + 2^{n - 1} \frac{1}{2^{n}})

= 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \dots + \frac{1}{2}

= 1 + \frac{n}{2}

but

{\frac{n}{2}}

is a divergent sequence. This shows that

S_{2^{n}}

is also divergent and hence the series

\sum_{n = 1}^{\infty} \frac{1}{n}

is divergent.

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