Maui1opj

2023-03-22

How to find the sum of the infinite geometric series 0.9 + 0.09 + 0.009 +…?

srnessgebf

Beginner2023-03-23Added 8 answers

For instance:

$\frac{1}{3}=0.\overline{3}$

$3\times \frac{1}{3}=0.\overline{9}$ but is also equal to 1, Consequently, both numbers are equal.

$\frac{1}{3}=0.\overline{3}$

$3\times \frac{1}{3}=0.\overline{9}$ but is also equal to 1, Consequently, both numbers are equal.

Drake Peters

Beginner2023-03-24Added 5 answers

The other response is accurate, but it's also important to note how using the geometric series formula would produce the outcome:

For $\left|r\right|<1$ and $a\ne 0$ we have $\sum _{n=0}^{\infty}{r}^{n}=\frac{a}{1-r}$

Applying this, as $\left|\frac{1}{10}\right|<1$, we have

$0.9+0.09+0.009+...=\sum _{n=0}^{\infty}\frac{9}{10}{\left(\frac{1}{10}\right)}^{n}$

$=\frac{\frac{9}{10}}{1-\frac{1}{10}}$

$=\frac{\frac{9}{10}}{\frac{9}{10}}$

$=1$

For $\left|r\right|<1$ and $a\ne 0$ we have $\sum _{n=0}^{\infty}{r}^{n}=\frac{a}{1-r}$

Applying this, as $\left|\frac{1}{10}\right|<1$, we have

$0.9+0.09+0.009+...=\sum _{n=0}^{\infty}\frac{9}{10}{\left(\frac{1}{10}\right)}^{n}$

$=\frac{\frac{9}{10}}{1-\frac{1}{10}}$

$=\frac{\frac{9}{10}}{\frac{9}{10}}$

$=1$

What is the derivative of the work function?

How to use implicit differentiation to find $\frac{dy}{dx}$ given $3{x}^{2}+3{y}^{2}=2$?

How to differentiate $y=\mathrm{log}{x}^{2}$?

The solution of a differential equation y′′+3y′+2y=0 is of the form

A) ${c}_{1}{e}^{x}+{c}_{2}{e}^{2x}$

B) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{3x}$

C) ${c}_{1}{e}^{-x}+{c}_{2}{e}^{-2x}$

D) ${c}_{1}{e}^{-2x}+{c}_{2}{2}^{-x}$How to find instantaneous velocity from a position vs. time graph?

How to implicitly differentiate $\sqrt{xy}=x-2y$?

What is 2xy differentiated implicitly?

How to find the sum of the infinite geometric series given $1+\frac{2}{3}+\frac{4}{9}+...$?

Look at this series: 1.5, 2.3, 3.1, 3.9, ... What number should come next?

A. 4.2

B. 4.4

C. 4.7

D. 5.1What is the derivative of $\frac{x+1}{y}$?

How to find the volume of a cone using an integral?

What is the surface area of the solid created by revolving $f\left(x\right)={e}^{2-x},x\in [1,2]$ around the x axis?

How to differentiate ${x}^{\frac{2}{3}}+{y}^{\frac{2}{3}}=4$?

The differential coefficient of $\mathrm{sec}\left({\mathrm{tan}}^{-1}\left(x\right)\right)$.

How to apply the ratio test to determine if $\Sigma \frac{n!}{{n}^{n}}$ from $n=[1,\infty )$ is convergent to divergent?