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(-1)^(n+1)/n
Sum of series/ (-1)^(n+1)/n

Sum of series (-1)^(n+1)/n


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The solution

You have entered [src] 
  oo           
____           
\   `          
 \        n + 1
  \   (-1)     
  /   ---------
 /        n    
/___,          
n = 1
∑n=1∞(−1)n+1n\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n} 
Sum((-1)^(n + 1)/n, (n, 1, oo))
The radius of convergence of the power series
Given number:
(−1)n+1n\frac{\left(-1\right)^{n + 1}}{n}
It is a series of species
an(cx−x0)dna_{n} \left(c x - x_{0}\right)^{d n}
- power series.
The radius of convergence of a power series can be calculated by the formula:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=(−1)n+1na_{n} = \frac{\left(-1\right)^{n + 1}}{n}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞(n+1n)1 = \lim_{n \to \infty}\left(\frac{n + 1}{n}\right)
Let's take the limit
we find
True

False
The rate of convergence of the power series
1.07.01.52.02.53.03.54.04.55.05.56.06.50.01.5
The answer [src] 
log(2)
log⁡(2)\log{\left(2 \right)} 
log(2)
Numerical answer [src] 
0.693147180559945309417232121458
0.693147180559945309417232121458
The graph
Sum of series (-1)^(n+1)/n

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