Mister Exam
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- Product of series
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How to use it?
- Sum of series:
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1/(n(n+2))
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n^3/2^n
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(2n-1)/2^n
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1/(4n^2-1)
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Identical expressions
- ((- one)^(n+ one)/(n!(n+ zero . five)))(two ^(-n- zero . five)- one)
- (( minus 1) to the power of (n plus 1) divide by (n!(n plus 0.5)))(2 to the power of ( minus n minus 0.5) minus 1)
- (( minus one) to the power of (n plus one) divide by (n!(n plus zero . five)))(two to the power of ( minus n minus zero . five) minus one)
- ((-1)(n+1)/(n!(n+0.5)))(2(-n-0.5)-1)
- -1n+1/n!n+0.52-n-0.5-1
- -1^n+1/n!n+0.52^-n-0.5-1
- ((-1)^(n+1) divide by (n!(n+0.5)))(2^(-n-0.5)-1)
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Similar expressions
- ((-1)^(n+1)/(n!(n+0.5)))(2^(n-0.5)-1)
- ((-1)^(n+1)/(n!(n+0.5)))(2^(-n-0.5)+1)
- ((-1)^(n+1)/(n!(n+0.5)))(2^(-n+0.5)-1)
- ((-1)^(n+1)/(n!(n-0.5)))(2^(-n-0.5)-1)
- ((1)^(n+1)/(n!(n+0.5)))(2^(-n-0.5)-1)
- ((-1)^(n-1)/(n!(n+0.5)))(2^(-n-0.5)-1)
Sum of series ((-1)^(n+1)/(n!(n+0.5)))(2^(-n-0.5)-1)
The solution
You have entered
[src]
oo ____ \ ` \ n + 1 \ (-1) / -n - 1/2 \ / ------------*\2 - 1/ / n!*(n + 1/2) /___, n = 0
$$\sum_{n=0}^{\infty} \frac{\left(-1\right)^{n + 1}}{\left(n + \frac{1}{2}\right) n!} \left(2^{- n - \frac{1}{2}} - 1\right)$$
Sum(((-1)^(n + 1)/((factorial(n)*(n + 1/2))))*(2^(-n - 1/2) - 1), (n, 0, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right)^{n + 1}}{\left(n + \frac{1}{2}\right) n!} \left(2^{- n - \frac{1}{2}} - 1\right)$$
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
$$a_{n} = \frac{\left(-1\right)^{n + 1} \left(2^{- n - \frac{1}{2}} - 1\right)}{\left(n + \frac{1}{2}\right) n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + \frac{3}{2}\right) \left|{\frac{\left(1 - 2^{- (n + \frac{1}{2})}\right) \left(n + 1\right)!}{\left(1 - 2^{- (n + \frac{3}{2})}\right) n!}}\right|}{n + \frac{1}{2}}\right)$$
Let's take the limit
we find
$$\frac{\left(-1\right)^{n + 1}}{\left(n + \frac{1}{2}\right) n!} \left(2^{- n - \frac{1}{2}} - 1\right)$$
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
$$a_{n} = \frac{\left(-1\right)^{n + 1} \left(2^{- n - \frac{1}{2}} - 1\right)}{\left(n + \frac{1}{2}\right) n!}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(n + \frac{3}{2}\right) \left|{\frac{\left(1 - 2^{- (n + \frac{1}{2})}\right) \left(n + 1\right)!}{\left(1 - 2^{- (n + \frac{3}{2})}\right) n!}}\right|}{n + \frac{1}{2}}\right)$$
Let's take the limit
we find
False
False
The rate of convergence of the power series
The answer
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/ ___\
____ ____ |\/ 2 |
\/ pi *erf(1) - \/ pi *erf|-----|
\ 2 /
$$- \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2}}{2} \right)} + \sqrt{\pi} \operatorname{erf}{\left(1 \right)}$$
sqrt(pi)*erf(1) - sqrt(pi)*erf(sqrt(2)/2)
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