Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
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How to use it?
- Sum of series:
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(-1)^(n-1)/ln(n+1)
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30
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1/ln^n(n+1)
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(-1)^n/(nlnn)
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Identical expressions
- (sqrt(n+ one))/(n^ five +2n+ one)
- ( square root of (n plus 1)) divide by (n to the power of 5 plus 2n plus 1)
- ( square root of (n plus one)) divide by (n to the power of five plus 2n plus one)
- (√(n+1))/(n^5+2n+1)
- (sqrt(n+1))/(n5+2n+1)
- sqrtn+1/n5+2n+1
- (sqrt(n+1))/(n⁵+2n+1)
- sqrtn+1/n^5+2n+1
- (sqrt(n+1)) divide by (n^5+2n+1)
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Similar expressions
- (sqrt(n-1))/(n^5+2n+1)
- (sqrt(n+1))/(n^5-2n+1)
- (sqrt(n+1))/(n^5+2n-1)
Sum of series (sqrt(n+1))/(n^5+2n+1)
The solution
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oo ____ \ ` \ _______ \ \/ n + 1 ) ------------ / 5 / n + 2*n + 1 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\sqrt{n + 1}}{\left(n^{5} + 2 n\right) + 1}$$
Sum(sqrt(n + 1)/(n^5 + 2*n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sqrt{n + 1}}{\left(n^{5} + 2 n\right) + 1}$$
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
$$a_{n} = \frac{\sqrt{n + 1}}{n^{5} + 2 n + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\sqrt{n + 1} \left(2 n + \left(n + 1\right)^{5} + 3\right)}{\sqrt{n + 2} \left(n^{5} + 2 n + 1\right)}\right)$$
Let's take the limit
we find
$$\frac{\sqrt{n + 1}}{\left(n^{5} + 2 n\right) + 1}$$
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
$$a_{n} = \frac{\sqrt{n + 1}}{n^{5} + 2 n + 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\sqrt{n + 1} \left(2 n + \left(n + 1\right)^{5} + 3\right)}{\sqrt{n + 2} \left(n^{5} + 2 n + 1\right)}\right)$$
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