Mister Exam
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- 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:
- a^n
-
(-1)^n*(2*2)^(2*n)/factorial(2*n)
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38
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1/n(n^2-1)
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Identical expressions
- one /(two *n+ one)!
- 1 divide by (2 multiply by n plus 1)!
- one divide by (two multiply by n plus one)!
- 1/(2n+1)!
- 1/2n+1!
- 1 divide by (2*n+1)!
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Similar expressions
- (-1)^n+1/(2n+1)!
- 1/(2*n-1)!
Sum of series 1/(2*n+1)!
The solution
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oo ___ \ ` \ 1 ) ---------- / (2*n + 1)! /__, n = 1
$$\sum_{n=1}^{\infty} \frac{1}{\left(2 n + 1\right)!}$$
Sum(1/factorial(2*n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{1}{\left(2 n + 1\right)!}$$
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
$$a_{n} = \frac{1}{\left(2 n + 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(2 n + 3\right)!}{\left(2 n + 1\right)!}}\right|$$
Let's take the limit
we find
$$\frac{1}{\left(2 n + 1\right)!}$$
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
$$a_{n} = \frac{1}{\left(2 n + 1\right)!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(2 n + 3\right)!}{\left(2 n + 1\right)!}}\right|$$
Let's take the limit
we find
False
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