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:
-
(2^n+3^n)/6^n
- cos(xi+yi)
-
(1/sqrt(n))-(1/sqrt(n+1))
-
n/9^n
-
Identical expressions
- (one /sqrt(n))-(one /sqrt(n+ one))
- (1 divide by square root of (n)) minus (1 divide by square root of (n plus 1))
- (one divide by square root of (n)) minus (one divide by square root of (n plus one))
- (1/√(n))-(1/√(n+1))
- 1/sqrtn-1/sqrtn+1
- (1 divide by sqrt(n))-(1 divide by sqrt(n+1))
-
Similar expressions
- (1/(sqrt(n)))-1/(sqrt(n+1))
- (1/sqrt(n))-(1/sqrt(n-1))
- (1/sqrt(n))+(1/sqrt(n+1))
- ((1/(sqrt(n)))-(1/(sqrt(n+1))))
- (1/sqrt(n)-1/sqrt(n+1))
Sum of series (1/sqrt(n))-(1/sqrt(n+1))
The solution
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oo ____ \ ` \ / 1 1 \ \ |----- - ---------| / | ___ _______| / \\/ n \/ n + 1 / /___, n = 1
$$\sum_{n=1}^{\infty} \left(- \frac{1}{\sqrt{n + 1}} + \frac{1}{\sqrt{n}}\right)$$
Sum(1/(sqrt(n)) - 1/sqrt(n + 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$- \frac{1}{\sqrt{n + 1}} + \frac{1}{\sqrt{n}}$$
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
$$a_{n} = - \frac{1}{\sqrt{n + 1}} + \frac{1}{\sqrt{n}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\frac{1}{\sqrt{n + 1}} - \frac{1}{\sqrt{n}}}{\frac{1}{\sqrt{n + 2}} - \frac{1}{\sqrt{n + 1}}}}\right|$$
Let's take the limit
we find
$$- \frac{1}{\sqrt{n + 1}} + \frac{1}{\sqrt{n}}$$
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
$$a_{n} = - \frac{1}{\sqrt{n + 1}} + \frac{1}{\sqrt{n}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\frac{1}{\sqrt{n + 1}} - \frac{1}{\sqrt{n}}}{\frac{1}{\sqrt{n + 2}} - \frac{1}{\sqrt{n + 1}}}}\right|$$
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