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:
-
n^2
-
(3^n+2^n)/6^n
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(4^(n+1)-10^n)/20^n
-
1/((3*n-2)*(3*n+1))
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Identical expressions
- (four ^(n+ one)- ten ^n)/ twenty ^n
- (4 to the power of (n plus 1) minus 10 to the power of n) divide by 20 to the power of n
- (four to the power of (n plus one) minus ten to the power of n) divide by twenty to the power of n
- (4(n+1)-10n)/20n
- 4n+1-10n/20n
- 4^n+1-10^n/20^n
- (4^(n+1)-10^n) divide by 20^n
-
Similar expressions
- (4^(n+1)+10^n)/20^n
- (4^(n-1)-10^n)/20^n
Sum of series (4^(n+1)-10^n)/20^n
The solution
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oo ____ \ ` \ n + 1 n \ 4 - 10 ) ------------ / n / 20 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{- 10^{n} + 4^{n + 1}}{20^{n}}$$
Sum((4^(n + 1) - 10^n)/20^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{- 10^{n} + 4^{n + 1}}{20^{n}}$$
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
$$a_{n} = - 10^{n} + 4^{n + 1}$$
and
$$x_{0} = -20$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-20 + \lim_{n \to \infty} \left|{\frac{10^{n} - 4^{n + 1}}{10^{n + 1} - 4^{n + 2}}}\right|\right)$$
Let's take the limit
we find
$$R = 0$$
$$\frac{- 10^{n} + 4^{n + 1}}{20^{n}}$$
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
$$a_{n} = - 10^{n} + 4^{n + 1}$$
and
$$x_{0} = -20$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-20 + \lim_{n \to \infty} \left|{\frac{10^{n} - 4^{n + 1}}{10^{n + 1} - 4^{n + 2}}}\right|\right)$$
Let's take the limit
we find
False
False
$$R = 0$$
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