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20 Cards in this Set
- Front
- Back
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if the sequence of partial sums has a limit L... |
the infinite series converges to that limit |
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if the sequence of partial sums diverges... |
the infinite series also diverges |
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Increasing |
if a_n+1 > an |
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nondecreasing |
if a(n+1) is greater than or equal to a(n) |
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decreasing |
a(n+1)< a(n) |
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non-increasing |
a(n+1)< or equal to a(n) |
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monotonic |
either nonincreasing or nondecreasing (moves in one direction) |
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bounded |
there is a number M such that |a(n)| < or equal to M for all relevant values of n |
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Geometric Sequences |
r^n or ar^n (each term multiplied by the previous term by a fixed constant, called ratio) r and a does not = 0 are real numbers |
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Squeeze Theorem |
for a(n) < or equal to b(n) < or equal to c(n) if a and c have the same limits then b has the same limit L |
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Growth Rate of Sequences |
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r^k from k=0 to infinity |
1/(1-r) if |r|<1 diverges if |r|> or equal to 1 |
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Divergence Test |
if lim a(n) as n approaches infinity, does not equal zero it diverges if not zero, it is inconclusive |
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"P" Series |
from 1- infinity--- 1/(k^p) diverges if p< or equal to 1 converges if p>1 |
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Integral Test |
for positive, decreasing for all n and continuous, decreasing f(x) a(n) converges if the limit of the integral from 1-infinity of f(x) converges------vice versa for divergence |
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Ratio Test |
r = lim k approaches infinity, a(k+1)/a(k) 0< or equal to r < 1, converges r>1, diverges r=1, inconclusive |
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Comparison Test |
a(k) and b(k) ----series with positive terms 0<a(k)<or equal to b(k) and b(k) converges, then a(k) converges 0<b(k)< or equal to a(k) and b(k) diverges, then a(k) diverges. |
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Harmonic Series |
1/k diverges even though the terms of the series approach 0 |
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R(n) |
a(k)-a(k+1) |
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Root Test |
p= lim (as k appr. infinity) of the kth root of a(k) if p<1 not 0, converges if p>1, diverges p=1, test is inconclusive |
