So since my last post, we have started to learn about sequences and series.
A sequence is a list of numbers defined by some equation, and a series is the addition or substraction of this list of numbers.
Converge vs. Diverge.
Sequences converge if they have a limit.
Sequence diverge if they don't have a limit.
If the limit of a sequence at infinity is infinity, then the sequence diverges.
If a sequence is bounded and monotonic* then it is converges.
If a sequence is bounded and not monotonic then the sequence diverges.
If a sequence is not bounded and monotonic then the sequence diverges.
*Monotonic - if terms are always increasing or always decreasing.
If a sequence of partial sums converge then the series converges.
If a sequence of partial sums diverge then the series diverges.
Series:
If something asks you to find the nth partial sum this means to find the sum at the nth term.
An arithmetic series will never converge. It will always diverge (as it approaches infinity).
Need to know:
Lim as n -> infinity (1 + (1/n))^n = e
Sequence properties follow limit properties (infinity limits at least).
Questions:
How do I use the squeeze theorem? I'm completely unsure of what to do.
How to find equations of series.
I'm somewhat unconfident of what to do anytime I see a sigma.
Ryan B.
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when you see a sigma, i'm pretty sure you take the limit as x goes to infinity. and if you get an infinity, it diverges. if you get a number, it converges.
ReplyDeleteWhen you see a sigma, read the directions! You may simply be finding the first five terms or partial sums. However if you want to know about divergence or convergence you need to take the limit!
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