Question

Use the ratio test to determine whether the series is convergent or divergent.

Answer 1

Answer:

Option B is correct, i.e. Divergent series.

Step-by-step explanation:

Given the series is:-

(2/1²) + (4/2²) + (8/3²) + (16/4²) +.....

n-th term of the series would be:- aₙ = (2ⁿ)/(n²)

(n+1)-th term of the series would be:- aₙ₊₁ = (2ⁿ⁺¹)/(n+1)²

Using Ratio test:-

$L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_{n}}|\\L = \lim_{n \to \infty} |\frac{(2^{n+1})/(n+1)^2}{(2^n)/(n^2)}|\\L = \lim_{n \to \infty} |\frac{2*n^2}{(n+1)^2}|\\L = \lim_{n \to \infty} |\frac{2}{(1 + 1/n)^2}|\\L = |\frac{2}{(1 + 1/\infty)^2}|\\L = |\frac{2}{(1 + 0)^2}|\\L = 2$

If L > 1, then series is divergent.

Since we got L = 2 and 2 > 1. It means given series is divergent.

Hence, option B is correct, i.e. Divergent series.

Answer 2

Answer:

Divergent

Step-by-step explanation:

Given series has general term as

$\frac{2^n}{n^2}$

n+1 th term =$\frac{2^(n+1)}{(n+1)^2}$

WE have to check whether this series converges or diverges

Let us use ratio test

Ratio of n+1 th term to nth term

=$\frac{2n^2}{(n+1)^2} =\frac{2}{(1+\frac{1}{n} }^2$

by dividing numerator and denominator by n square

Take limits as n tends to infinity

The ratio tends to 2

Since ratio >2, the series diverges