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:
-
sqrt((n+4)/((n^4)+4))
-
(7^n-3^n)/21^n
-
sin(pi/2^(n-1))
- sin(n*x)/n^2
-
Identical expressions
- sin(pi/ two ^(n- one))
- sinus of ( Pi divide by 2 to the power of (n minus 1))
- sinus of ( Pi divide by two to the power of (n minus one))
- sin(pi/2(n-1))
- sinpi/2n-1
- sinpi/2^n-1
- sin(pi divide by 2^(n-1))
-
Similar expressions
- sin(pi/2^(n+1))
Sum of series sin(pi/2^(n-1))
The solution
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oo ____ \ ` \ / pi \ \ sin|------| / | n - 1| / \2 / /___, n = 1
$$\sum_{n=1}^{\infty} \sin{\left(\frac{\pi}{2^{n - 1}} \right)}$$
Sum(sin(pi/2^(n - 1)), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\sin{\left(\frac{\pi}{2^{n - 1}} \right)}$$
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
$$a_{n} = \sin{\left(2^{1 - n} \pi \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(2^{1 - n} \pi \right)}}{\sin{\left(2^{- n} \pi \right)}}}\right|$$
Let's take the limit
we find
$$\sin{\left(\frac{\pi}{2^{n - 1}} \right)}$$
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
$$a_{n} = \sin{\left(2^{1 - n} \pi \right)}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\sin{\left(2^{1 - n} \pi \right)}}{\sin{\left(2^{- n} \pi \right)}}}\right|$$
Let's take the limit
we find
False
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