Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
-
How to use it?
- Sum of series:
- (((-1)^(n))*(16*x)^(n))/n!
-
(-1)^n/(n+1)
- (3×x^n)/(4n^2-9)
-
(100!)/(n!*(100-n)!)*((0.0125)^n)*((1-0.0125)^(100-n))
-
Identical expressions
- (- one)^n/(n+ one)
- ( minus 1) to the power of n divide by (n plus 1)
- ( minus one) to the power of n divide by (n plus one)
- (-1)n/(n+1)
- -1n/n+1
- -1^n/n+1
- (-1)^n divide by (n+1)
-
Similar expressions
- (-1)^n/(n-1)
- (1)^n/(n+1)
Sum of series (-1)^n/(n+1)
The solution
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oo ____ \ ` \ n \ (-1) / ----- / n + 1 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n}}{n + 1}$$
Sum((-1)^n/(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right)^{n}}{n + 1}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \frac{1}{n + 1}$$
and
$$x_{0} = 1$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(1 + \lim_{n \to \infty}\left(\frac{n + 2}{n + 1}\right)\right)$$
Let's take the limit
we find
$$\frac{\left(-1\right)^{n}}{n + 1}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \frac{1}{n + 1}$$
and
$$x_{0} = 1$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(1 + \lim_{n \to \infty}\left(\frac{n + 2}{n + 1}\right)\right)$$
Let's take the limit
we find
False
The rate of convergence of the power series
The graph
Examples of finding the sum of a series
- The Sum of the Power Series
x^n/n
(x-1)^n
- Factorial
1/2^(n!)
n^2/n!
x^n/n!
k!/(n!*(n+k)!)
- Flint Hills Series
csc(n)^2/n^3
- Basel problem
1/n^2
1/n^4
1/n^6
- Harmonic series
1/n
- Grandi's series
(-1)^n
- Alternating series
(-1)^(n + 1)/n
(n + 2)*(-1)^(n - 1)
(3*n - 1)/(-5)^n
(-1)^(n - 1)*n/(6*n - 5)
- Newton–Mercator series
(-1)^(n + 1)/n*x^n
- Exam the series for convergence
(3*n - 1)/(-5)^n