Mister Exam
Other calculators
- Taylor Series Step by Step
- Fourier series step by step
- Differential equations Step by Step
- Product of series
-
How to use it?
- Sum of series:
-
(3^n-1)/factorial(n)
- n*x^n
-
sin(1/n)/sqrt(n)
-
ln(n/10)/(exp(n/10)+1)
-
Identical expressions
- sin(one /n)/sqrt(n)
- sinus of (1 divide by n) divide by square root of (n)
- sinus of (one divide by n) divide by square root of (n)
- sin(1/n)/√(n)
- sin1/n/sqrtn
- sin(1 divide by n) divide by sqrt(n)
-
Similar expressions
- arcsin(1/n)/(sqrt(n+1)-sqrt(n))
Sum of series sin(1/n)/sqrt(n)
The solution
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oo _____ \ ` \ /1\ \ sin|-| \ \n/ / ------ / ___ / \/ n /____, n = 1
$$\sum_{n=1}^{\infty} \frac{\sin{\left(\frac{1}{n} \right)}}{\sqrt{n}}$$
Sum(sin(1/n)/sqrt(n), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\sin{\left(\frac{1}{n} \right)}}{\sqrt{n}}$$
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
$$a_{n} = \frac{\sin{\left(\frac{1}{n} \right)}}{\sqrt{n}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\sqrt{n + 1} \left|{\frac{\sin{\left(\frac{1}{n} \right)}}{\sin{\left(\frac{1}{n + 1} \right)}}}\right|}{\sqrt{n}}\right)$$
Let's take the limit
we find
$$\frac{\sin{\left(\frac{1}{n} \right)}}{\sqrt{n}}$$
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
$$a_{n} = \frac{\sin{\left(\frac{1}{n} \right)}}{\sqrt{n}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\sqrt{n + 1} \left|{\frac{\sin{\left(\frac{1}{n} \right)}}{\sin{\left(\frac{1}{n + 1} \right)}}}\right|}{\sqrt{n}}\right)$$
Let's take the limit
we find
True
False
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