Mister Exam
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- 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:
-
n*2^n
-
n^3
- t^k*(-1)^k*a^(2*k)*cos(a*x)*e^(a*z)/factorial(k)
-
(2n+1)/(n(n+2)(n+4))
-
Identical expressions
- sin(one hundred *pi*n)/n
- sinus of (100 multiply by Pi multiply by n) divide by n
- sinus of (one hundred multiply by Pi multiply by n) divide by n
- sin(100pin)/n
- sin100pin/n
- sin(100*pi*n) divide by n
Sum of series sin(100*pi*n)/n
The solution
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oo ___ \ ` \ sin(100*pi*n) ) ------------- / n /__, n = 1
$$\sum_{n=1}^{\infty} \frac{\sin{\left(100 \pi n \right)}}{n}$$
Sum(sin((100*pi)*n)/n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sin{\left(100 \pi n \right)}}{n}$$
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
$$a_{n} = \frac{\sin{\left(100 \pi n \right)}}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left|{\frac{\sin{\left(100 \pi n \right)}}{\sin{\left(\pi \left(100 n + 100\right) \right)}}}\right|}{n}\right)$$
Let's take the limit
we find
$$\frac{\sin{\left(100 \pi n \right)}}{n}$$
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
$$a_{n} = \frac{\sin{\left(100 \pi n \right)}}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + 1\right) \left|{\frac{\sin{\left(100 \pi n \right)}}{\sin{\left(\pi \left(100 n + 100\right) \right)}}}\right|}{n}\right)$$
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