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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