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Terms & Definitions of Improper Integrals

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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
Any sequence that is monotonic and bounded will ...
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