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:
-
1/(3n-2)(3n+1)
-
n*3^n+1
-
0.5^n
- z^(n+1)/factorial(n+1)
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Identical expressions
- ln(one +(one /n*sqrt(n)))+ one /n
- ln(1 plus (1 divide by n multiply by square root of (n))) plus 1 divide by n
- ln(one plus (one divide by n multiply by square root of (n))) plus one divide by n
- ln(1+(1/n*√(n)))+1/n
- ln(1+(1/nsqrt(n)))+1/n
- ln1+1/nsqrtn+1/n
- ln(1+(1 divide by n*sqrt(n)))+1 divide by n
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Similar expressions
- ln(1-(1/n*sqrt(n)))+1/n
- ln(1+(1/n*sqrt(n)))-1/n
Sum of series ln(1+(1/n*sqrt(n)))+1/n
The solution
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[src]
oo ____ \ ` \ / / ___\ \ \ | | \/ n | 1| / |log|1 + -----| + -| / \ \ n / n/ /___, n = 1
$$\sum_{n=1}^{\infty} \left(\log{\left(\frac{\sqrt{n}}{n} + 1 \right)} + \frac{1}{n}\right)$$
Sum(log(1 + sqrt(n)/n) + 1/n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\log{\left(\frac{\sqrt{n}}{n} + 1 \right)} + \frac{1}{n}$$
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
$$a_{n} = \log{\left(1 + \frac{1}{\sqrt{n}} \right)} + \frac{1}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\log{\left(1 + \frac{1}{\sqrt{n}} \right)} + \frac{1}{n}}{\log{\left(1 + \frac{1}{\sqrt{n + 1}} \right)} + \frac{1}{n + 1}}\right)$$
Let's take the limit
we find
$$\log{\left(\frac{\sqrt{n}}{n} + 1 \right)} + \frac{1}{n}$$
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
$$a_{n} = \log{\left(1 + \frac{1}{\sqrt{n}} \right)} + \frac{1}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\log{\left(1 + \frac{1}{\sqrt{n}} \right)} + \frac{1}{n}}{\log{\left(1 + \frac{1}{\sqrt{n + 1}} \right)} + \frac{1}{n + 1}}\right)$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
oo ____ \ ` \ /1 / 1 \\ \ |- + log|1 + -----|| / |n | ___|| / \ \ \/ n // /___, n = 1
$$\sum_{n=1}^{\infty} \left(\log{\left(1 + \frac{1}{\sqrt{n}} \right)} + \frac{1}{n}\right)$$
Sum(1/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