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:
-
(-1)^n
-
1/((2n-1)(2n+1))
-
sqrt(n^3+2)/(n^2+3)
-
cos(pi*n/180)
-
Identical expressions
- pi^n+ one / two ^2n
- Pi to the power of n plus 1 divide by 2 squared n
- Pi to the power of n plus one divide by two squared n
- pin+1/22n
- pi^n+1/2²n
- pi to the power of n+1/2 to the power of 2n
- pi^n+1 divide by 2^2n
-
Similar expressions
- pi^n-1/2^2n
Sum of series pi^n+1/2^2n
The solution
You have entered
[src]
oo ____ \ ` \ / n 1 \ \ |pi + --*n| / | 2 | / \ 2 / /___, n = 1
$$\sum_{n=1}^{\infty} \left(\pi^{n} + \frac{n}{4}\right)$$
Sum(pi^n + (1/2)^2*n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\pi^{n} + \frac{n}{4}$$
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
$$a_{n} = \pi^{n} + \frac{n}{4}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\pi^{n} + \frac{n}{4}}{\pi^{n + 1} + \frac{n}{4} + \frac{1}{4}}\right)$$
Let's take the limit
we find
$$\pi^{n} + \frac{n}{4}$$
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
$$a_{n} = \pi^{n} + \frac{n}{4}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\pi^{n} + \frac{n}{4}}{\pi^{n + 1} + \frac{n}{4} + \frac{1}{4}}\right)$$
Let's take the limit
we find
False
False
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