Question

This is a miscellaneous series and we have to find its sum.

$\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + ...$
Both Numerator and denominator are in AP here.

Kindly solve​

Answer 1

SOLUTION

TO DETERMINE

The sum of the series

$\displaystyle \sf{ \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + ... \: .. }$

EVALUATION

Here the given series is

$\displaystyle \sf{ \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + ... \: .. }$

First we check whether the series is convergent or divergent

The given series is of the form

$\displaystyle \sf{ \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + ... \: .. }$

$\displaystyle \sf{ = \sum\limits_{n=1}^{ \infty } \: \frac{n}{n + 1} }$

$\displaystyle \sf{ = \sum\limits_{n=1}^{ \infty } a_n \: \: \: \: where \: \: \: a_n = \frac{n}{n + 1} }$

Now

$\displaystyle \sf{ \lim_{n \to \infty } \: a_n}$

$\displaystyle \sf{ = \lim_{n \to \infty } \: \frac{n}{n + 1} }$

$\displaystyle \sf{ = \lim_{n \to \infty } \: \frac{1}{1+ \frac{1}{n} } }$

$\displaystyle \sf{ = \frac{1}{1+ 0} }$

$\displaystyle \sf{ = 1 \: \ne \: 0 }$

Hence the given series is divergent

So the sum of the divergent series can not be determined or tends off to infinity.

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

$1.$ If integral of (x^-3)5^(1/x^2)dx=k5^(1/x^2), then k is equal to?

https://brainly.in/question/15427882

2. If integral of (x^-3)5^(1/x^2)dx=k5^(1/x^2), then k is equal to?

https://brainly.in/question/15427882