Mister Exam
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- Taylor Series Step by Step
- Fourier series step by step
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- Product of series
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How to use it?
- Sum of series:
-
((-1)^(n+1))/n
-
sin(n)
-
(-1)^n*sqrt(n)/(n+100)
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(0.15/0.05)^1.3
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Identical expressions
- (sin(n)/(n))-((sin(n+ one))/(n+ one))
- ( sinus of (n) divide by (n)) minus (( sinus of (n plus 1)) divide by (n plus 1))
- ( sinus of (n) divide by (n)) minus (( sinus of (n plus one)) divide by (n plus one))
- sinn/n-sinn+1/n+1
- (sin(n) divide by (n))-((sin(n+1)) divide by (n+1))
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Similar expressions
- (sin(n)/(n))+((sin(n+1))/(n+1))
- ((sin(n))/(n))-((sin(n+1))/(n+1))
- (sin(n)/(n))-((sin(n-1))/(n+1))
- (sin(n)/(n))-((sin(n+1))/(n-1))
Sum of series (sin(n)/(n))-((sin(n+1))/(n+1))
The solution
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oo ___ \ ` \ /sin(n) sin(n + 1)\ ) |------ - ----------| / \ n n + 1 / /__, n = 1
$$\sum_{n=1}^{\infty} \left(- \frac{\sin{\left(n + 1 \right)}}{n + 1} + \frac{\sin{\left(n \right)}}{n}\right)$$
Sum(sin(n)/n - sin(n + 1)/(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$- \frac{\sin{\left(n + 1 \right)}}{n + 1} + \frac{\sin{\left(n \right)}}{n}$$
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
$$a_{n} = - \frac{\sin{\left(n + 1 \right)}}{n + 1} + \frac{\sin{\left(n \right)}}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\frac{\sin{\left(n + 1 \right)}}{n + 1} - \frac{\sin{\left(n \right)}}{n}}{\frac{\sin{\left(n + 2 \right)}}{n + 2} - \frac{\sin{\left(n + 1 \right)}}{n + 1}}}\right|$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty} \left|{\frac{\frac{\sin{\left(n + 1 \right)}}{n + 1} - \frac{\sin{\left(n \right)}}{n}}{\frac{\sin{\left(n + 2 \right)}}{n + 2} - \frac{\sin{\left(n + 1 \right)}}{n + 1}}}\right|$$
$$- \frac{\sin{\left(n + 1 \right)}}{n + 1} + \frac{\sin{\left(n \right)}}{n}$$
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
$$a_{n} = - \frac{\sin{\left(n + 1 \right)}}{n + 1} + \frac{\sin{\left(n \right)}}{n}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\frac{\sin{\left(n + 1 \right)}}{n + 1} - \frac{\sin{\left(n \right)}}{n}}{\frac{\sin{\left(n + 2 \right)}}{n + 2} - \frac{\sin{\left(n + 1 \right)}}{n + 1}}}\right|$$
Let's take the limit
we find
$$1 = \lim_{n \to \infty} \left|{\frac{\frac{\sin{\left(n + 1 \right)}}{n + 1} - \frac{\sin{\left(n \right)}}{n}}{\frac{\sin{\left(n + 2 \right)}}{n + 2} - \frac{\sin{\left(n + 1 \right)}}{n + 1}}}\right|$$
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