Solveeit Logo
Login

Question

Question: The sum of the series\(\frac{1}{1.2} + \frac{1.3}{1.2.3.4} + \frac{1.3.5}{1.2.3.4.5.6} + .....\infty...

The sum of the series11.2+1.31.2.3.4+1.3.51.2.3.4.5.6+.....∞\frac{1}{1.2} + \frac{1.3}{1.2.3.4} + \frac{1.3.5}{1.2.3.4.5.6} + .....\infty is.

A

15e15e

B

e1/2+ee^{1/2} + e

C

e1/2−1e^{1/2} - 1

D

e1/2−ee^{1/2} - e

Answer

e1/2−1e^{1/2} - 1

Explanation

Solution

The nth term of given series is

log⁡e2\log_{e}2

Tn=1⋅2⋅3⋅4….(2n−2)(2n−1)(2n)1⋅2⋅3⋅4…(2n−1)(2n)T _ { n } = \frac { 1 \cdot 2 \cdot 3 \cdot 4 \ldots . ( 2 n - 2 ) ( 2 n - 1 ) ( 2 n ) } { 1 \cdot 2 \cdot 3 \cdot 4 \ldots ( 2 n - 1 ) ( 2 n ) } log⁡e(23)\log_{e}(2\sqrt{3})

TnT _ { n } log⁡e2\log_{e}2,log⁡e(23)\log_{e}\left( \frac{2}{\sqrt{3}} \right)