+0

# Evaluate the infinite geometric series: ​

0
54
3
+397

Evaluate the infinite geometric series:$$1-\frac{2}{7}+\frac{4}{49}-\frac{8}{343}+\dots$$

Apr 26, 2020

#1
0

sumfor(n, 1, 100,  (-1)^(n + 1)* (7/2)^(1 - n) = it converges to  7 / 9

Apr 26, 2020
#2
+24995
+3

Evaluate the infinite geometric series: $$1-\dfrac{2}{7}+\dfrac{4}{49}-\dfrac{8}{343}+\dots$$

$$\begin{array}{|rcll|} \hline && \mathbf{1-\dfrac{2}{7}+\dfrac{4}{49}-\dfrac{8}{343}+\ldots} \\\\ &=& 1+\left(-\dfrac{2}{7}\right) +\left(-\dfrac{2}{7}\right)^2 +\left(-\dfrac{2}{7}\right)^3 +\ldots \quad | \quad \text{ratio}\ = -\dfrac{2}{7} \\ \hline \end{array}$$

Formula:

$$\begin{array}{|rcll|} \hline \mathbf{s} &=& \mathbf{\dfrac{1}{1-r}} \quad | \quad r=\text{ratio},\ s=\text{infinite sum} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline \mathbf{s} &=& \mathbf{\dfrac{1}{1-r}} \quad | \quad r = -\dfrac{2}{7} \\\\ s &=& \dfrac{1}{1-\left(-\dfrac{2}{7}\right)} \\\\ s &=& \dfrac{1}{1+ \dfrac{2}{7} } \\\\ s &=& \dfrac{1}{\dfrac{7+2}{7} } \\\\ s &=& \dfrac{7} {7+2} \\\\ \mathbf{s} &=& \mathbf{\dfrac{7}{9}} \\ \hline \end{array}$$

Apr 27, 2020
#3
+633
+1

Here is the proof for the formula.

AnExtremelyLongName  Apr 27, 2020