Mister Exam

Other calculators

((-1)^(k+1))/(3^k)
Sum of series/ ((-1)^(k+1))/(3^k)

Sum of series ((-1)^(k+1))/(3^k)


∑
=

The solution

You have entered [src] 
  oo           
____           
\   `          
 \        k + 1
  \   (-1)     
   )  ---------
  /        k   
 /        3    
/___,          
k = 1
$$\sum_{k=1}^{\infty} \frac{\left(-1\right)^{k + 1}}{3^{k}}$$ 
Sum((-1)^(k + 1)/3^k, (k, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right)^{k + 1}}{3^{k}}$$
It is a series of species
$$a_{k} \left(c x - x_{0}\right)^{d k}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{k \to \infty} \left|{\frac{a_{k}}{a_{k + 1}}}\right|}{c}$$
In this case
$$a_{k} = \left(-1\right)^{k + 1}$$
and
$$x_{0} = -3$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-3 + \lim_{k \to \infty} 1\right)$$
Let's take the limit
we find
False

$$R = 0$$
The rate of convergence of the power series
The answer [src] 
1/4
$$\frac{1}{4}$$ 
1/4
Numerical answer [src] 
0.250000000000000000000000000000
0.250000000000000000000000000000
The graph
Sum of series ((-1)^(k+1))/(3^k)

    Examples of finding the sum of a series

    © Mister Exam – Calculator