Question

1/1.2.3+3/2.3.4+5/3.4.5+... Test the convergence

Answer 1

Answer:

everything can change in the world

Answer 2

Answer:

$\sum \frac{2n + 1}{n(n + 1)(n + 2)}$

The series is convergent

Step-by-step explanation:

1) Find the nth term of the series.

note that in the numerator the sequence is 1,3,5..

so the nth term will be 2n+1

And in the denominator the sequence is 1.2.3, 2.3.4,...

so the nth term will be n(n+1)(n+2)

so the nth term of given series Σan is given as

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

2) Consider another series with nth term

$b_n = \frac{1}{ {n}^{2} }$

3) Find the ratio of nth term of both the series

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

$\frac{a_n}{b_n} = \frac{ {n}^{3}(2 + \frac{1}{n} )}{ {n}^{3} (1 + \frac{1}{n} )(1 + \frac{2}{n}) }$

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

4) Find it's limit say l

$l = \lim_{n \to \infty } \frac{a_n}{b_n} = \frac{2}{(1)(1)} = 2$

$\lim_{n \to \infty } \frac{1}{n} = 0$

5) Using the test that if l is not equal to zero or infinity then both the series behave same, here as the limit is 2 both the series will behave same.

6) We know that series Σbn is convergent by p-test as p= 2 >1

7) Therefore the the series Σan will also be convergent.

Therefore the Series

$\sum \frac{2n + 1}{n(n + 1)(n + 2)}$ is convergent.