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/ln(5^n)
-
ln(5)^n/n!
-
(-1)^(n+1)/ln(n+2)
-
((n^2)^1/3)*atan(1/n^2)
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Identical expressions
- (- one)^(n+ one)/ln(n+ two)
- ( minus 1) to the power of (n plus 1) divide by ln(n plus 2)
- ( minus one) to the power of (n plus one) divide by ln(n plus two)
- (-1)(n+1)/ln(n+2)
- -1n+1/lnn+2
- -1^n+1/lnn+2
- (-1)^(n+1) divide by ln(n+2)
-
Similar expressions
- (-1)^(n-1)/ln(n+2)
- (1)^(n+1)/ln(n+2)
- (-1)^(n+1)/ln(n-2)
Sum of series (-1)^(n+1)/ln(n+2)
The solution
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oo ____ \ ` \ n + 1 \ (-1) / ---------- / log(n + 2) /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{\log{\left(n + 2 \right)}}$$
Sum((-1)^(n + 1)/log(n + 2), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right)^{n + 1}}{\log{\left(n + 2 \right)}}$$
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
$$a_{n} = \frac{\left(-1\right)^{n + 1}}{\log{\left(n + 2 \right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\log{\left(n + 3 \right)}}{\log{\left(n + 2 \right)}}\right)$$
Let's take the limit
we find
$$\frac{\left(-1\right)^{n + 1}}{\log{\left(n + 2 \right)}}$$
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
$$a_{n} = \frac{\left(-1\right)^{n + 1}}{\log{\left(n + 2 \right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\log{\left(n + 3 \right)}}{\log{\left(n + 2 \right)}}\right)$$
Let's take the limit
we find
True
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