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:
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24/((5-3n)(2-3n))
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(5^(1/sqrtn)-1)^3
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arctan(n+3)-arctan(n+1)
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1/n(n^2-1)
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Identical expressions
- (five ^(one /sqrtn)- one)^ three
- (5 to the power of (1 divide by square root of n) minus 1) cubed
- (five to the power of (one divide by square root of n) minus one) to the power of three
- (5^(1/√n)-1)^3
- (5(1/sqrtn)-1)3
- 51/sqrtn-13
- (5^(1/sqrtn)-1)³
- (5 to the power of (1/sqrtn)-1) to the power of 3
- 5^1/sqrtn-1^3
- (5^(1 divide by sqrtn)-1)^3
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Similar expressions
- (5^(1/sqrtn)+1)^3
Sum of series (5^(1/sqrtn)-1)^3
The solution
You have entered
[src]
oo _____ \ ` \ 3 \ / 1 \ \ | ----- | / | ___ | / | \/ n | / \5 - 1/ /____, n = 1
$$\sum_{n=1}^{\infty} \left(5^{\frac{1}{\sqrt{n}}} - 1\right)^{3}$$
Sum((5^(1/(sqrt(n))) - 1)^3, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(5^{\frac{1}{\sqrt{n}}} - 1\right)^{3}$$
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
$$a_{n} = \left(5^{\frac{1}{\sqrt{n}}} - 1\right)^{3}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(5^{\frac{1}{\sqrt{n}}} - 1\right)^{2} \left|{5^{\frac{1}{\sqrt{n}}} - 1}\right| \left|{\frac{1}{\left(5^{\frac{1}{\sqrt{n + 1}}} - 1\right)^{3}}}\right|\right)$$
Let's take the limit
we find
$$\left(5^{\frac{1}{\sqrt{n}}} - 1\right)^{3}$$
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
$$a_{n} = \left(5^{\frac{1}{\sqrt{n}}} - 1\right)^{3}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\left(5^{\frac{1}{\sqrt{n}}} - 1\right)^{2} \left|{5^{\frac{1}{\sqrt{n}}} - 1}\right| \left|{\frac{1}{\left(5^{\frac{1}{\sqrt{n + 1}}} - 1\right)^{3}}}\right|\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
oo _____ \ ` \ 3 \ / 1 \ \ | -----| / | ___| / | \/ n | / \-1 + 5 / /____, n = 1
$$\sum_{n=1}^{\infty} \left(5^{\frac{1}{\sqrt{n}}} - 1\right)^{3}$$
Sum((-1 + 5^(1/sqrt(n)))^3, (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