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What is an Infinite Series?
The infinite sum of a1+a2+a3+...
The Limit of a Series exist when?
There is a sum of the Series that exist Sn = (a/(1-r))
The Limit of a Series DNE when?
The sum is not known or is infinite.
Converges to the value of the Sum
Therefore if a Sum of the Series exist, the Series?
Diverges to Infinity
Therefor if a Sum of the Series DNE, the Series?
a - the first term
In the Geometric series, the key variables involved are?
The Geometric Series Converges if?
The absolute value of the Common Ratio is = 0 or is Less than +1
The Geometric Series Diverges if?
The value of the Common Ratio is less than/equal to -1 and greater than/equal to +1
Geometric Series Converges
When r is 0 or less than 1?
Geometric Series Diverges to Infinity
When r is greater than 1?
Geometric Series Diverges
When r is less than -1?
When r is equal to -1?
Geometric Series DNE (= 0, when n is even) and (= a, when n is odd)
What is the key variable to a P Series?
When the value of P is greater than 0.
What happens when P=1 in a P Series?
It is known as the Harmonic (P) Series
The P Series Converges when?
P is greater than 1
The P Series Diverges when?
P is less than/equal to 1 and is greater than 0
Telescoping Series Converges if?
The Limit of the sequence of bn+1 exists
The Telescoping Series Diverges
If the limit of the sequence of bn+1 doesn't exist?
What is a Convergent Series?
The Sum or Difference of Two Converging Series
It does not change the behavior of the Series
What is the effect of a Constant Multiple on a Series?
In the Divergence test, the Series Diverges when?
The limit of the sequence an (of the Series) does not equal to 0
The Divergence Test Fails, either converges or diverges
If the limit of the sequence an does equal to zero?
The satisfaction of the 3 Conditions
The Main idea for the Integral Test to work on a Series is?
The First Condition (Positive) must involve:
f(x) to be greater than 0 and x to be greater than 1
The Second Condition (Continuous) must involve:
f(x) is greater for all x greater than or equal to 1
The Third Condition (Decreasing) must involve:
f'(x) is less than 0 and x is greater than or equal to 1
The Integral Series Converges when?
The Limit of the Integral of f(x) Exists as a constant (IT)
The Integral Series Diverges when?
The Limit of the Integral of f(x) is equal to Positive/Negative Infinity (IT)
The Ratio Test Converges when?
The Limit of the Series is greater than or equal to 0 and is less than 1 (RT)
The Ratio Test Diverges when?
The Limit of the Series is less than 1 or equal to positive infinity (RT)
The Ratio Test Fails when?
The Limit of the Series is equal to 1 (RT)
The nth Root Test Converges when?
The Limit of the Series is greater than or equal to 0 and is less than 1 (nRT)
The nth Root Test Diverges when?
The Limit of the Series is less than 1 or equal to positive infinity (nRT)
The Root Test Fails when?
The Limit of the Series is equal to 1 (nRT)
The Comparison Test Converges when?
an is less/equal to bn, as n is greater/equal to 1
The Comparison Test Diverges when?
an is greater/equal to bn, as n is greater/equal to 1
The Comparison Test Fails when?
an is less/equal to bn, but Series bn Diverges
Both Series of an and bn Converge
The Limit Comparison Test when L = 0?
Both Series of an and bn Diverge
The Limit Comparison Test when L = Infinity?
-1 < x < 1
notation for the interval of convergence. (1 is a placeholder)
x^2 < 1
notation for the ratio of convergence. (1 is a placeholder)
a/1-r
the formula to find the sum (Sn) of the first n terms of a finite geometric series