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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(2^n+(-1)^n)/5^n
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(n+1)^2/2^(n-1)
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3/(n*(n+2))
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1/(4n^2-1)
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Identical expressions
- one /n- two /(n)^(one / two)+log(one + one /(n)^(one / two))
- 1 divide by n minus 2 divide by (n) to the power of (1 divide by 2) plus logarithm of (1 plus 1 divide by (n) to the power of (1 divide by 2))
- one divide by n minus two divide by (n) to the power of (one divide by two) plus logarithm of (one plus one divide by (n) to the power of (one divide by two))
- 1/n-2/(n)(1/2)+log(1+1/(n)(1/2))
- 1/n-2/n1/2+log1+1/n1/2
- 1/n-2/n^1/2+log1+1/n^1/2
- 1 divide by n-2 divide by (n)^(1 divide by 2)+log(1+1 divide by (n)^(1 divide by 2))
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Similar expressions
- 1/n-2/(n)^(1/2)+log(1-1/(n)^(1/2))
- 1/n+2/(n)^(1/2)+log(1+1/(n)^(1/2))
- 1/n-2/(n)^(1/2)-log(1+1/(n)^(1/2))
Sum of series 1/n-2/(n)^(1/2)+log(1+1/(n)^(1/2))
The solution
You have entered
[src]
oo ____ \ ` \ /1 2 / 1 \\ \ |- - ----- + log|1 + -----|| / |n ___ | ___|| / \ \/ n \ \/ n // /___, n = 1
$$\sum_{n=1}^{\infty} \left(\left(\frac{1}{n} - \frac{2}{\sqrt{n}}\right) + \log{\left(1 + \frac{1}{\sqrt{n}} \right)}\right)$$
Sum(1/n - 2/sqrt(n) + log(1 + 1/(sqrt(n))), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\left(\frac{1}{n} - \frac{2}{\sqrt{n}}\right) + \log{\left(1 + \frac{1}{\sqrt{n}} \right)}$$
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
$$a_{n} = \log{\left(1 + \frac{1}{\sqrt{n}} \right)} + \frac{1}{n} - \frac{2}{\sqrt{n}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\log{\left(1 + \frac{1}{\sqrt{n}} \right)} + \frac{1}{n} - \frac{2}{\sqrt{n}}}{\log{\left(1 + \frac{1}{\sqrt{n + 1}} \right)} + \frac{1}{n + 1} - \frac{2}{\sqrt{n + 1}}}}\right|$$
Let's take the limit
we find
$$\left(\frac{1}{n} - \frac{2}{\sqrt{n}}\right) + \log{\left(1 + \frac{1}{\sqrt{n}} \right)}$$
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
$$a_{n} = \log{\left(1 + \frac{1}{\sqrt{n}} \right)} + \frac{1}{n} - \frac{2}{\sqrt{n}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\log{\left(1 + \frac{1}{\sqrt{n}} \right)} + \frac{1}{n} - \frac{2}{\sqrt{n}}}{\log{\left(1 + \frac{1}{\sqrt{n + 1}} \right)} + \frac{1}{n + 1} - \frac{2}{\sqrt{n + 1}}}}\right|$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
oo ____ \ ` \ /1 2 / 1 \\ \ |- - ----- + log|1 + -----|| / |n ___ | ___|| / \ \/ n \ \/ n // /___, n = 1
$$\sum_{n=1}^{\infty} \left(\log{\left(1 + \frac{1}{\sqrt{n}} \right)} + \frac{1}{n} - \frac{2}{\sqrt{n}}\right)$$
Sum(1/n - 2/sqrt(n) + log(1 + 1/sqrt(n)), (n, 1, oo))
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