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Converges or Diverges
21
Aaahh!!! Real Series
Divergent or Convergent?
Series
Is the sum of an ordered list of elements.
Let´s Check!
Geometric
- Constant ratio between succesive terms
Geometric
Condition
Examples
Divergence test
The divergence test can determine whether a series diverges, and if it does diverge, it can't possibly converge.
Divergence test
A series will diverge if...
Divergence
does not exist or if
If L = 0
IF
The series converges, but we can't conclude to which value it will converge.
Tips
- The divergence test will usually be the first test of whether or not a series converges.
TIPS
Some examples of series that can be solved by the Divergence Test are:
Examples
Example
For more explanation you can watch:
http://xaktly.com/DivergenceTest.html
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Harmonic Series
if the general term An has the next form:
Harmonic
Why does these series diverge?
Diverges!
Tadaaah! Check here.....
https://mathwithbaddrawings.com/2015/07/15/the-strange-music-of-the-harmonic-series/
Example
Telescopic sum
- whose partial sums only have a fixed number of terms after cancellation
Telescopic sum
1. an must be split into the minimum two parts
- With polynomials, by factoring the denominator we can split an through partial fractions
Steps
Partial fractions
1. Factor the denominator
2. Place the factors on the denominators
3. Propose the numerators of the fractions as one degree less than the denominators degree
4. Solve for A,B, .... Multiply both sides by the common denominator
5. Solve the system of equations
6. Plug in the values of A,B, .. in the preposition of partial fraction
Examples
Ratio test
This test is helpful to determine if a series is absolutely convergent.
Ratio test
It will converge if...
Convergence
It will diverge if...
And it will be inconclusive if...
AND
Tips
TIPS
- If there is a factorial, go directly to this test.
- If there is an exponential do this test.
- If L = 1 in the Root Test, don’t try the Ratio Test because it will be 1.
Absolutely convergent
A series an is called absolutely convergent if the series of
absolute values an is convergent.
Convergence
Some examples of series that can be solved by the Ratio Test are:
Examples
Example
For more explanation you can watch...
https://www.youtube.com/watch?v=-Md4CEW0-zM&feature=youtu.be&t=474
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Root Test
When it looks like...
Root test
Root test!
Diverges or converges?
Example
Tip
Use this method when you have (F(x))^n
https://www.youtube.com/watch?v=a43dldQ5Mvo
Useful Link
Team Members
María Cristina del Carmen Inda
Gisela Yasmin González García
Andrea Carolina Venegas Sánchez