Recent questions tagged infinite-series

​​​​​The sum of the following infinite series is:$$ \frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\cdots $$$\pi$$1+e$$e-1$$e$

The value of the series $1+\sin x+\cos ^{2} x+\sin ^{3} x+\cdots$ at $x=\frac{\pi}{4}$ is $\_\_\_\_\_\_\_$.$\frac{1}{\sqrt{2}+1}$$\frac{\sqrt{2}}{\sqrt{2}+1}$$\frac{1}{\s...

An infinite series $S$ is given as:$S=1+2/3+3/9+4/27+5/81+\:.\dots$ (to infinity)The value of $S$ is ______________________ (round off to $2$ decimal places).

Which of the following are geometric series?$1,\:6,\:11,\:16,\:21,\:26,\:\dots$$9,\:6,\:3,\:0,\:-3,\:-6,\:\dots$$1,\:3,\:9,\:27,\:81,\:\dots$$4,\:-8,\:16,\:-32,\:64,\:\do...

Consider the following infinite series:$$1+ r+r^2 +r^3+ \dots \dots \infty$$ If $r = 0.3$, then the sum of this infinite series is ____________

If $1+r + r^2+ r^3 +\dots \infty = 1.5$, then, $1 + 2r + 3r^2 + 4r^3 + \dots \infty = $ (up to two dcimal places) ________