Infinite Series Overview - Seton Hall University



Infinite Series Overview

If [pic] is an infinite sequence, then [pic] is an infinite series. It can converge or diverge. If [pic] is an infinite series, define the N-th partial sum [pic]. Then the series [pic] is said to converge if the sequence of partial sums[pic] converges, and to diverge if that sequence diverges. A series is said to converge absolutely if [pic] converges, and conditionally convergent if it does converge but not absolutely (if the series contains only positive term, absolute convergence and “regular” convergence is the same).

Special Series

Geometric Series: [pic]

• If [pic] the series converges to [pic]

• If [pic] the series diverges

Telescoping Series: a series where subsequent terms cancel each other out.

p-Series: [pic]

• If [pic] the series converges

• If [pic] the series diverges

Note: The p-series for [pic] is also called Harmonic Series (which diverges)

It is relatively easy to tell whether a series converges or diverges, but with the exception of the Geometric Series and Telescoping Series it is usually difficult to find out what the actual limit is.

Convergence Tests

Divergence Test: If [pic] then [pic]diverges

Limit Comparison Test: If [pic] and [pic] are two sequences such that [pic] exists and is not zero, then the two series [pic] and [pic] are comparable, i.e. they either both converge or both diverge.

Ratio Test: Consider the series [pic] and compute [pic]

• If [pic] the series[pic]converges absolutely

• If [pic] the series [pic] diverges

• If [pic] there is no information

Alternating Series Test: Consider the series [pic]. If (i) the sequence [pic] is positive, (ii) [pic] is decreasing, and (iii) [pic] then the series converges.

Other, less frequently used converge tests include the (direct) Comparison Test, the Integral Test, and the Root Test.

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