Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
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- Product of series
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How to use it?
- Sum of series:
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1/(n(n+2))
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(2n-1)/2^n
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sqrt(n)-2*sqrt(n+1)+sqrt(n+2)
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log((n^2+1)/n^2)
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Identical expressions
- sin^ two (n+ one)/((n+ one)*sqrt(n+ one))
- sinus of squared (n plus 1) divide by ((n plus 1) multiply by square root of (n plus 1))
- sinus of to the power of two (n plus one) divide by ((n plus one) multiply by square root of (n plus one))
- sin^2(n+1)/((n+1)*√(n+1))
- sin2(n+1)/((n+1)*sqrt(n+1))
- sin2n+1/n+1*sqrtn+1
- sin²(n+1)/((n+1)*sqrt(n+1))
- sin to the power of 2(n+1)/((n+1)*sqrt(n+1))
- sin^2(n+1)/((n+1)sqrt(n+1))
- sin2(n+1)/((n+1)sqrt(n+1))
- sin2n+1/n+1sqrtn+1
- sin^2n+1/n+1sqrtn+1
- sin^2(n+1) divide by ((n+1)*sqrt(n+1))
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Similar expressions
- sin^2(n-1)/((n+1)*sqrt(n+1))
- sin^2(n+1)/((n-1)*sqrt(n+1))
- sin^2(n+1)/((n+1)*sqrt(n-1))
Sum of series sin^2(n+1)/((n+1)*sqrt(n+1))
The solution
You have entered
[src]
oo ____ \ ` \ 2 \ sin (n + 1) ) ----------------- / _______ / (n + 1)*\/ n + 1 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\sin^{2}{\left(n + 1 \right)}}{\sqrt{n + 1} \left(n + 1\right)}$$
Sum(sin(n + 1)^2/(((n + 1)*sqrt(n + 1))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sin^{2}{\left(n + 1 \right)}}{\sqrt{n + 1} \left(n + 1\right)}$$
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
$$a_{n} = \frac{\sin^{2}{\left(n + 1 \right)}}{\left(n + 1\right)^{\frac{3}{2}}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 2\right)^{\frac{3}{2}} \sin^{2}{\left(n + 1 \right)} \left|{\frac{1}{\sin^{2}{\left(n + 2 \right)}}}\right|}{\left(n + 1\right)^{\frac{3}{2}}}\right)$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 2\right)^{\frac{3}{2}} \sin^{2}{\left(n + 1 \right)} \left|{\frac{1}{\sin^{2}{\left(n + 2 \right)}}}\right|}{\left(n + 1\right)^{\frac{3}{2}}}\right)$$
$$\frac{\sin^{2}{\left(n + 1 \right)}}{\sqrt{n + 1} \left(n + 1\right)}$$
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
$$a_{n} = \frac{\sin^{2}{\left(n + 1 \right)}}{\left(n + 1\right)^{\frac{3}{2}}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 2\right)^{\frac{3}{2}} \sin^{2}{\left(n + 1 \right)} \left|{\frac{1}{\sin^{2}{\left(n + 2 \right)}}}\right|}{\left(n + 1\right)^{\frac{3}{2}}}\right)$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 2\right)^{\frac{3}{2}} \sin^{2}{\left(n + 1 \right)} \left|{\frac{1}{\sin^{2}{\left(n + 2 \right)}}}\right|}{\left(n + 1\right)^{\frac{3}{2}}}\right)$$
False
The rate of convergence of the power series
The answer
[src]
oo ____ \ ` \ 2 \ sin (1 + n) ) ----------- / 3/2 / (1 + n) /___, n = 1
$$\sum_{n=1}^{\infty} \frac{\sin^{2}{\left(n + 1 \right)}}{\left(n + 1\right)^{\frac{3}{2}}}$$
Sum(sin(1 + n)^2/(1 + n)^(3/2), (n, 1, oo))
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