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:
-
2^(2*n-1)/9^n
- k*x+b
- cos(n*p)/4
-
1/(2n+1)*2^2n+1
-
Identical expressions
- tg(one /nsqrtn)
- tg(1 divide by n square root of n)
- tg(one divide by n square root of n)
- tg(1/n√n)
- tg1/nsqrtn
- tg(1 divide by nsqrtn)
-
Similar expressions
- tg1/(n*sqrt(n))
Sum of series tg(1/nsqrtn)
The solution
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[src]
oo ____ \ ` \ / ___\ \ |\/ n | / tan|-----| / \ n / /___, n = 2
$$\sum_{n=2}^{\infty} \tan{\left(\frac{\sqrt{n}}{n} \right)}$$
Sum(tan(sqrt(n)/n), (n, 2, oo))
The radius of convergence of the power series
Given number:
$$\tan{\left(\frac{\sqrt{n}}{n} \right)}$$
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
$$a_{n} = \tan{\left(\frac{1}{\sqrt{n}} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\tan{\left(\frac{1}{\sqrt{n}} \right)}}{\tan{\left(\frac{1}{\sqrt{n + 1}} \right)}}}\right|$$
Let's take the limit
we find
$$\tan{\left(\frac{\sqrt{n}}{n} \right)}$$
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
$$a_{n} = \tan{\left(\frac{1}{\sqrt{n}} \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\tan{\left(\frac{1}{\sqrt{n}} \right)}}{\tan{\left(\frac{1}{\sqrt{n + 1}} \right)}}}\right|$$
Let's take the limit
we find
True
False
The rate of convergence of the power series
The answer
[src]
oo ____ \ ` \ / 1 \ \ tan|-----| / | ___| / \\/ n / /___, n = 2
$$\sum_{n=2}^{\infty} \tan{\left(\frac{1}{\sqrt{n}} \right)}$$
Sum(tan(1/sqrt(n)), (n, 2, 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