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:
- (((-1)^(n))*(16*x)^(n))/n!
-
(-1)^n/(n+1)
- (3×x^n)/(4n^2-9)
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(100!)/(n!*(100-n)!)*((0.0125)^n)*((1-0.0125)^(100-n))
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Identical expressions
- (three ×x^n)/(4n^ two - nine)
- (3×x to the power of n) divide by (4n squared minus 9)
- (three ×x to the power of n) divide by (4n to the power of two minus nine)
- (3×xn)/(4n2-9)
- 3×xn/4n2-9
- (3×x^n)/(4n²-9)
- (3×x to the power of n)/(4n to the power of 2-9)
- 3×x^n/4n^2-9
- (3×x^n) divide by (4n^2-9)
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Similar expressions
- (3×x^n)/(4n^2+9)
Sum of series (3×x^n)/(4n^2-9)
The solution
You have entered
[src]
oo ____ \ ` \ n \ 3*x ) -------- / 2 / 4*n - 9 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{3 x^{n}}{4 n^{2} - 9}$$
Sum((3*x^n)/(4*n^2 - 9), (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{3 x^{n}}{4 n^{2} - 9}$$
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
$$a_{n} = \frac{3}{4 n^{2} - 9}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty}\left(3 \left|{\frac{\frac{4 \left(n + 1\right)^{2}}{3} - 3}{4 n^{2} - 9}}\right|\right)$$
Let's take the limit
we find
$$R = 1$$
$$\frac{3 x^{n}}{4 n^{2} - 9}$$
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
$$a_{n} = \frac{3}{4 n^{2} - 9}$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty}\left(3 \left|{\frac{\frac{4 \left(n + 1\right)^{2}}{3} - 3}{4 n^{2} - 9}}\right|\right)$$
Let's take the limit
we find
$$R = 1$$
The answer
[src]
// / 2 \ \ || | 5 5*x 5*x | | || |- - - --- + ---- / 3\ / ___\| | || | 6 18 6 \5 - 5*x /*atanh\\/ x /| | ||-x*|---------------- + -----------------------| | || | 2 5/2 | | || \ x 6*x / | ||------------------------------------------------ for |x| <= 1| || 5 | || | 3*|< oo | || ____ | || \ ` | || \ n | || \ x | || ) --------- otherwise | || / 2 | || / -9 + 4*n | || /___, | || n = 1 | \\ /
$$3 \left(\begin{cases} - \frac{x \left(\frac{\frac{5 x^{2}}{6} - \frac{5 x}{18} - \frac{5}{6}}{x^{2}} + \frac{\left(5 - 5 x^{3}\right) \operatorname{atanh}{\left(\sqrt{x} \right)}}{6 x^{\frac{5}{2}}}\right)}{5} & \text{for}\: \left|{x}\right| \leq 1 \\\sum_{n=1}^{\infty} \frac{x^{n}}{4 n^{2} - 9} & \text{otherwise} \end{cases}\right)$$
3*Piecewise((-x*((-5/6 - 5*x/18 + 5*x^2/6)/x^2 + (5 - 5*x^3)*atanh(sqrt(x))/(6*x^(5/2)))/5, |x| <= 1), (Sum(x^n/(-9 + 4*n^2), (n, 1, oo)), True))
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