# Find the sum of the infinite geometric series. sum_{i=1}^infty12(-0.7)^{i-1}

Question
Series
Find the sum of the infinite geometric series.
$$\sum_{i=1}^\infty12(-0.7)^{i-1}$$

2021-02-10
To find:
The sum of the infinite series $$\sum_{i=1}^\infty12(-0.7)^{i-1}$$
Concept used:
The sum of infinite geometric series can be obtained by the formula,
$$S_\infty=\frac{a}{1-r}\quad(r<1)$$</span>
Here, a is first term, r is common ratio and $$S_\infty$$ is sum of infinite geometric series.
Calculation:
Expand the summation as follows:
$$\sum_{i=1}^\infty12(-0.7)^{i-1}=12(-0.7)^{1-1}+12(-0.7)^{2-1}+12(-0.7)^{3-1}+...$$
$$=12+12(-0.7)+12(-0.7)^2+...$$
It is observed from the series that the first term is 12 and common ratio is (−0.7) and common ratio is less than 1.
Substitute 12 for a and (−0.7) for r in equation (1).
$$S_{\infty}=\frac{12}{10(-0.7)}$$
$$=\frac{12}{1+0.7}$$
$$=\frac{12}{1.7}$$
$$=7.06$$
Thus, the sum of the infinite series is 7.06.

### Relevant Questions

Differentiate the power series for $$f(x)=xe^x$$. Use the result to find the sum of the infinite series
$$\sum_{n=0}^\infty\frac{n+1}{n!}$$
Differentiate the power series for $$\displaystyle{f{{\left({x}\right)}}}={x}{e}^{{x}}$$. Use the result to find the sum of the infinite series
$$\displaystyle{\sum_{{{n}={0}}}^{\infty}}{\frac{{{n}+{1}}}{{{n}!}}}$$
Consider the following infinite series.
a.Find the first four partial sums $$S_1,S_2,S_3,$$ and $$S_4$$ of the series.
b.Find a formula for the nth partial sum $$S_n$$ of the indinite series.Use this formula to find the next four partial sums $$S_5,S_6,S_7$$ and $$S_8$$ of the infinite series.
c.Make a conjecture for the value of the series.
$$\sum_{k=1}^\infty\frac{2}{(2k-1)(2k+1)}$$
Use the formula for the sum of a geometric series to find the sum, or state that the series diverges.
$$\sum_{n=0}^\infty\frac{3(-2)^n-5^n}{8^n}$$
Find the values of x for which the given geometric series converges. Also, find the sum of the series (as a function of x) for those values of x.
a) Find the values of x for which the given geometric series converges.
b) Find the sum of the series
$$\sum_{n=0}^\infty(-\frac12)^n(x-5)^n$$
a) Find the Maclaurin series for the function
$$f(x)=\frac11+x$$
b) Use differentiation of power series and the result of part a) to find the Maclaurin series for the function
$$g(x)=\frac{1}{(x+1)^2}$$
c) Use differentiation of power series and the result of part b) to find the Maclaurin series for the function
$$h(x)=\frac{1}{(x+1)^3}$$
d) Find the sum of the series
$$\sum_{n=3}^\infty \frac{n(n-1)}{2n}$$
This is a Taylor series problem, I understand parts a - c but I do not understand how to do part d where the answer is $$\frac72$$
$$\sum_{n=4}^\infty(-\frac49)^n$$
$$\sum_{n=2}^\infty\frac{7\cdot(-3)^n}{5^n}$$
$$\sum_{n=2}^\infty\frac{5^n}{12^n}$$
Evaluating an infinite series $$\sum_{k=1}^\infty(\frac{4}{3^k}-\frac{4}{3^{k+1}})$$ two ways.