Mister Exam
Other calculators
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- Product of series
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How to use it?
- Sum of series:
-
1
-
sin2n
- 2(x-6)^(4n)/(n(16)^n)
- nx^n/(8n-9)*3^n
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Identical expressions
- ((- one)^(n- one))*(five ^n+ two)*x^n
- (( minus 1) to the power of (n minus 1)) multiply by (5 to the power of n plus 2) multiply by x to the power of n
- (( minus one) to the power of (n minus one)) multiply by (five to the power of n plus two) multiply by x to the power of n
- ((-1)(n-1))*(5n+2)*xn
- -1n-1*5n+2*xn
- ((-1)^(n-1))(5^n+2)x^n
- ((-1)(n-1))(5n+2)xn
- -1n-15n+2xn
- -1^n-15^n+2x^n
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Similar expressions
- ((-1)^(n-1))*(5^n-2)*x^n
- ((1)^(n-1))*(5^n+2)*x^n
- ((-1)^(n+1))*(5^n+2)*x^n
Sum of series ((-1)^(n-1))*(5^n+2)*x^n
The solution
You have entered
[src]
oo ___ \ ` \ n - 1 / n \ n / (-1) *\5 + 2/*x /__, n = 1
$$\sum_{n=1}^{\infty} x^{n} \left(-1\right)^{n - 1} \left(5^{n} + 2\right)$$
Sum(((-1)^(n - 1)*(5^n + 2))*x^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$x^{n} \left(-1\right)^{n - 1} \left(5^{n} + 2\right)$$
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
$$a_{n} = \left(-1\right)^{n - 1} \left(5^{n} + 2\right)$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty}\left(\frac{5^{n} + 2}{5^{n + 1} + 2}\right)$$
Let's take the limit
we find
$$R = \frac{1}{5}$$
$$R = 0.2$$
$$x^{n} \left(-1\right)^{n - 1} \left(5^{n} + 2\right)$$
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
$$a_{n} = \left(-1\right)^{n - 1} \left(5^{n} + 2\right)$$
and
$$x_{0} = 0$$
,
$$d = 1$$
,
$$c = 1$$
then
$$R = \lim_{n \to \infty}\left(\frac{5^{n} + 2}{5^{n + 1} + 2}\right)$$
Let's take the limit
we find
$$R = \frac{1}{5}$$
$$R = 0.2$$
The answer
[src]
// -5*x \ // -x \ || ------- for 5*|x| < 1| || ----- for |x| < 1| || 1 + 5*x | || 1 + x | || | || | || oo | || oo | - |< ___ | - 2*|< ___ | || \ ` | || \ ` | || \ n n n | || \ n n | || / (-1) *5 *x otherwise | || / (-1) *x otherwise | || /__, | || /__, | \\n = 1 / \\n = 1 /
$$- 2 \left(\begin{cases} - \frac{x}{x + 1} & \text{for}\: \left|{x}\right| < 1 \\\sum_{n=1}^{\infty} \left(-1\right)^{n} x^{n} & \text{otherwise} \end{cases}\right) - \begin{cases} - \frac{5 x}{5 x + 1} & \text{for}\: 5 \left|{x}\right| < 1 \\\sum_{n=1}^{\infty} \left(-1\right)^{n} 5^{n} x^{n} & \text{otherwise} \end{cases}$$
-Piecewise((-5*x/(1 + 5*x), 5*|x| < 1), (Sum((-1)^n*5^n*x^n, (n, 1, oo)), True)) - 2*Piecewise((-x/(1 + x), |x| < 1), (Sum((-1)^n*x^n, (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