Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
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- Product of series
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How to use it?
- Sum of series:
-
n^(1/n)
-
5^n
-
2(1/(n^2+5n+6))
- ((-1)^n)*((x^n)/(2n*n!))
-
Identical expressions
- one / seven + one / twenty-seven + one /(2n- one)*(2n+ five)
- 1 divide by 7 plus 1 divide by 27 plus 1 divide by (2n minus 1) multiply by (2n plus 5)
- one divide by seven plus one divide by twenty minus seven plus one divide by (2n minus one) multiply by (2n plus five)
- 1/7+1/27+1/(2n-1)(2n+5)
- 1/7+1/27+1/2n-12n+5
- 1 divide by 7+1 divide by 27+1 divide by (2n-1)*(2n+5)
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Similar expressions
- 1/7-1/27+1/(2n-1)*(2n+5)
- 1/7+1/27-1/(2n-1)*(2n+5)
- 1/7+1/27+1/((2n-1)*(2n+5))
- 1/7+1/27+1/(2n-1)*(2n-5)
- 1/7+1/27+1/(2n+1)*(2n+5)
Sum of series 1/7+1/27+1/(2n-1)*(2n+5)
The solution
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oo ___ \ ` \ / 2*n + 5\ ) |1/7 + 1/27 + -------| / \ 2*n - 1/ /__, n = 1
$$\sum_{n=1}^{\infty} \left(\left(\frac{1}{27} + \frac{1}{7}\right) + \frac{2 n + 5}{2 n - 1}\right)$$
Sum(1/7 + 1/27 + (2*n + 5)/(2*n - 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(\frac{1}{27} + \frac{1}{7}\right) + \frac{2 n + 5}{2 n - 1}$$
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
$$a_{n} = \frac{34}{189} + \frac{2 n + 5}{2 n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\frac{34}{189} + \frac{2 n + 5}{2 n - 1}}\right|}{\frac{34}{189} + \frac{2 n + 7}{2 n + 1}}\right)$$
Let's take the limit
we find
$$\left(\frac{1}{27} + \frac{1}{7}\right) + \frac{2 n + 5}{2 n - 1}$$
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
$$a_{n} = \frac{34}{189} + \frac{2 n + 5}{2 n - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left|{\frac{34}{189} + \frac{2 n + 5}{2 n - 1}}\right|}{\frac{34}{189} + \frac{2 n + 7}{2 n + 1}}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
Numerical answer
The series diverges
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