Question

Test the
Convergence of series 2/1+2.5.8/1.5.9+2.5.8.11/1.5.9.13+....​

Answer 1

Answer:

Step-by-step explanation:

The series is convergent:-

Answer 2

The limit is less than 1, by the Ratio Test, the series converges absolutely.

the given series is convergent.

The given series is:

2/1 + 2.5.8/1.5.9 + 2.5.8.11/1.5.9.13 + ...

To test the convergence of this series, we will use the Ratio Test.

Ratio Test:

The series converges absolutely if the limit of the ratio of the (n+1)th term to the nth term as n approaches infinity is smaller than 1.

The series diverges if the limit is greater than 1 or infinite.

The test is not convincing if the limit is equal to 1.

Let a_n be the nth term of the series. Then,

$a_n = [2.5.8...(3n-1)]/[1.5.9...(4n-3)]$

$a_n+1 = [2.5.8...(3n+2)]/[1.5.9...(4n+1)]$

So, the ratio of the (n+1)th term to the nth term is:

$a_n+1/a_n = [(3n+2)/(4n+1)] * [(4n-3)/(3n-1)]$

$= [3(4n-3)/(4n+1)] * [(4n-3)/(3n-1)]$

$= [9(4n-3)/(4n+1)(3n-1)]$

As n approaches infinity, the limit of this expression is:

lim(n→∞) [9(4n-3)/(4n+1)(3n-1)] = 9/12 = 3/4

Since the limit is less than 1, by the Ratio Test, the series converges absolutely.

Therefore, the given series is convergent.

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https://brainly.in/question/32296921

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