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

Bulk modulus of water is $2 \times 10^9 \ N/m^2$. The pressure required to increase the volume of water by $0.1 \%$ in $N/m^2$ is:

For the stationary wave $y = 4 \sin \left( \frac{\pi x}{15} \right) \cos (96 \pi t)$,the distance between a node and the next antinode is

An elevator weighing $6000\, kg$ is pulled upward by a cable with an acceleration of $5\, m/s^2$. Taking $g$ to be $10\, m/s^2$,then the tension in the cable is ........... $N$.

Which of the following is a hydride?

What is the molecular formula of the product formed when benzene is reacted with ethyl chloride in the presence of anhydrous aluminium chloride?

Vedclass Test Series

Exam Paper Generator

Online Exam Module