Mister Exam
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- Taylor Series Step by Step
- Fourier series step by step
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- Product of series
-
How to use it?
- Sum of series:
-
sqrt((n+4)/((n^4)+4))
-
(7^n-3^n)/21^n
-
sin(pi/2^(n-1))
- sin(n*x)/n^2
- Derivative of:
- (7^n-3^n)/21^n
-
Identical expressions
- (seven ^n- three ^n)/ twenty-one ^n
- (7 to the power of n minus 3 to the power of n) divide by 21 to the power of n
- (seven to the power of n minus three to the power of n) divide by twenty minus one to the power of n
- (7n-3n)/21n
- 7n-3n/21n
- 7^n-3^n/21^n
- (7^n-3^n) divide by 21^n
-
Similar expressions
- (7^n+3^n)/21^n
Sum of series (7^n-3^n)/21^n
The solution
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oo ____ \ ` \ n n \ 7 - 3 ) ------- / n / 21 /___, n = 1
$$\sum_{n=1}^{\infty} \frac{- 3^{n} + 7^{n}}{21^{n}}$$
Sum((7^n - 3^n)/21^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{- 3^{n} + 7^{n}}{21^{n}}$$
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
$$a_{n} = - 3^{n} + 7^{n}$$
and
$$x_{0} = -21$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-21 + \lim_{n \to \infty} \left|{\frac{3^{n} - 7^{n}}{3^{n + 1} - 7^{n + 1}}}\right|\right)$$
Let's take the limit
we find
$$R = 0$$
$$\frac{- 3^{n} + 7^{n}}{21^{n}}$$
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
$$a_{n} = - 3^{n} + 7^{n}$$
and
$$x_{0} = -21$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-21 + \lim_{n \to \infty} \left|{\frac{3^{n} - 7^{n}}{3^{n + 1} - 7^{n + 1}}}\right|\right)$$
Let's take the limit
we find
False
False
$$R = 0$$
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