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:
-
(3^n-1)/6^n
-
(5^n-2^n)/10^n
-
1/(3n+2)(3n+1)
-
(factorial(n-1)/factorial(n+1))^(n*(n+1))
-
Identical expressions
- (five ^n- two ^n)/ ten ^n
- (5 to the power of n minus 2 to the power of n) divide by 10 to the power of n
- (five to the power of n minus two to the power of n) divide by ten to the power of n
- (5n-2n)/10n
- 5n-2n/10n
- 5^n-2^n/10^n
- (5^n-2^n) divide by 10^n
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Similar expressions
- (5^n+2^n)/10^n
Sum of series (5^n-2^n)/10^n
The solution
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oo ____ \ ` \ n n \ 5 - 2 ) ------- / n / 10 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{- 2^{n} + 5^{n}}{10^{n}}$$
Sum((5^n - 2^n)/10^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{- 2^{n} + 5^{n}}{10^{n}}$$
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
$$a_{n} = - 2^{n} + 5^{n}$$
and
$$x_{0} = -10$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-10 + \lim_{n \to \infty} \left|{\frac{2^{n} - 5^{n}}{2^{n + 1} - 5^{n + 1}}}\right|\right)$$
Let's take the limit
we find
$$R = 0$$
$$\frac{- 2^{n} + 5^{n}}{10^{n}}$$
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
$$a_{n} = - 2^{n} + 5^{n}$$
and
$$x_{0} = -10$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-10 + \lim_{n \to \infty} \left|{\frac{2^{n} - 5^{n}}{2^{n + 1} - 5^{n + 1}}}\right|\right)$$
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