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:
-
1/n*(n+1)
-
7+k
-
6/(3n-1)(3n+5)
-
6/((3*n-1)*(3*n+5))
-
Identical expressions
- ((- one)^(n+ one))/(ln(n+ one))
- (( minus 1) to the power of (n plus 1)) divide by (ln(n plus 1))
- (( minus one) to the power of (n plus one)) divide by (ln(n plus one))
- ((-1)(n+1))/(ln(n+1))
- -1n+1/lnn+1
- -1^n+1/lnn+1
- ((-1)^(n+1)) divide by (ln(n+1))
-
Similar expressions
- ((-1)^(n-1))/(ln(n+1))
- ((-1)^(n+1))/(ln(n-1))
- ((1)^(n+1))/(ln(n+1))
Sum of series ((-1)^(n+1))/(ln(n+1))
The solution
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oo ____ \ ` \ n + 1 \ (-1) / ---------- / log(n + 1) /___, n = 2
$$\sum_{n=2}^{\infty} \frac{\left(-1\right)^{n + 1}}{\log{\left(n + 1 \right)}}$$
Sum((-1)^(n + 1)/log(n + 1), (n, 2, oo))
The radius of convergence of the power series
Given number:
$$\frac{\left(-1\right)^{n + 1}}{\log{\left(n + 1 \right)}}$$
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
$$a_{n} = \frac{\left(-1\right)^{n + 1}}{\log{\left(n + 1 \right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\log{\left(n + 2 \right)}}{\log{\left(n + 1 \right)}}\right)$$
Let's take the limit
we find
$$\frac{\left(-1\right)^{n + 1}}{\log{\left(n + 1 \right)}}$$
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
$$a_{n} = \frac{\left(-1\right)^{n + 1}}{\log{\left(n + 1 \right)}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\log{\left(n + 2 \right)}}{\log{\left(n + 1 \right)}}\right)$$
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