1+frac{1+2}{2 !}+frac{1+2+2^2}{3 !}+ldots is | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The general term $T_n$ of the series is given by $T_n = \frac{1+2+2^2+\ldots+2^{n-1}}{n !}$.Using the sum of a geometric progression,$T_n = \frac{

Vedclass · Accepted Answer

$e^2-e$

Vedclass · Answer

(D) The general term $T_n$ of the series is given by $T_n = \frac{1+2+2^2+\ldots+2^{n-1}}{n !}$.Using the sum of a geometric progression,$T_n = \frac{2^n-1}{n !} = \frac{2^n}{n !} - \frac{1}{n !}$.The sum of the series is $S = \sum_{n=1}^{\infty} T_n = \sum_{n=1}^{\infty} \frac{2^n}{n !} - \sum_{n=1}^{\infty} \frac{1}{n !}$.We know that $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n !} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \ldots$.Thus,$\sum_{n=1}^{\infty} \frac{2^n}{n !} = e^2 - 1$ and $\sum_{n=1}^{\infty} \frac{1}{n !} = e - 1$.Therefore,$S = (e^2 - 1) - (e - 1) = e^2 - e$.

$1+\frac{1+2}{2 !}+\frac{1+2+2^2}{3 !}+\ldots$ is equal to

Similar Questions

Which of the following is not correct regarding the properties of ionic compounds?

The perimeter of a sector is a constant. If its area is to be maximum,the sectorial angle is:

Two coherent light sources having intensity in the ratio $2x$ produce an interference pattern. The ratio $\frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}}$ will be

In a gene bank,valuable plant materials are stored in a living state. $A$ gene bank is a part of which of the following?

Sertoli cells are regulated by the pituitary hormone known as:

Vedclass Test Series

Exam Paper Generator

Online Exam Module