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L`Hospital's Rule

If direct substitution of the limit as x approaches a f(x)/g(x) yields an indeterminate form of 0/0 or infinity/infinity then, limit x-> a f(x)/g(x) = limit as x -> a f`(x)/g`(x)

Determinate forms are

0/n = 0, n/0 = infinity , infinity + infinity = infinity, 1*infinity = infinity , infinity^1 = infinity

Make indeterminate power forms to look like ..

0/0 or infinity/infinity, so that L`Hospitals Rule can be used

With Indeterminate products, use

If limit x -> a f(x)* g(x) yields 0 (+-)infinity, change to f(x) / 1(g(x)

Indeterminate powers, f(x)^g(x) yields 0^0, infinity^0 , 1^infinity, then use..

natural log (ln) by moving the exponent down in front of ln

A proper definite integral is ..

The integral from a to b, f(x) dx is equal to f(b) - f(a) if f`(x) is continuous on [a,b]

Improper integrals include ...

infinite intervals or infinite discontinuous over a closed interval

Sequence is a...

list generated by some rule. The list has terms a-sub1, a-sub2, a-sub3, .... denotes by {a-sub1, a-sub2, a-sub3, ...}

Series is a ..

sum made up of the terms in a sequence list

limit as n approaches infinity a-subn = L means

as n approaches infinity the terms a-subn are getting arbitrarily closer to L

the sequence converges to L

If a sequence has a limit L , then we say

the sequence diverges

If a sequence does not have a limit, then

An increasing sequence is ..

one of the form A(n+1) > A(n) Ex: A(n)=n/n+1 an increasing sequence

A decreasing sequence is ..

one of the form A(n+1) < A(n) /ex A(n)=1/n is a decreasing sequence

A monotonic sequence is ...

a sequence that either only increases or decreases

A bounded above sequence is

if the terms A(n) </= M Ex: A(n)=n/(n+1) is bounded above by 1

A bounded below sequence is ...

if the terms A(n) >/= M Ex A(n)=n is bounded below by 1

A bounded sequence is ..

a sequence that is bounded below and above A(n)=(-1)^n is bounded below and above by 1 and -1

converge

Any sequence that is monotonic and bounded will ...

S

If a series has a sum, the sum is denoted by

S(n) is a ...

partial sum ( Sum of the first n terms)

the series(sum) will diverge

If the sequence(list) of partial sums {Sn} diverges then

the series(sum) will converge. ( A limit exists)

If the sequence(list) of partial sums {Sn} converges then

limit n -> infinity {Sn}

To determine if a sequence converges, consider

The sum of a series (S) is ..

the limit of the sequence of partial sums