Sequences And Series Flashcards
Arithmetic series sum
n/2(2a+(n-1)d))
Finding the nth term of a geometric sequence
Geometric Sequence
A sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number.
An=a1*r^(n-1)
Geometric Series
Adding up a geometric sequence.
S=a1(1-r^n)/(1-r). This is the sum of any finite geometric series.
Geometric series sum for infinite series if r is between 0 and 1
With the formula s=a1(1-r^n)/(1-r) if r is bigger then 1, then it will sum up to infinity. But if r is between 0 and 1, then the sum will converge because the numbers will then get so small that they turn into zero.
Geometric series where n is to infinity.
With the equation a1*(1-r^n)/(1-r). The r^n will turn to zero if r is less then 1 because the number will get so small
Geometric series for an infinite series
If the r is less then 1, then the formula turns into series=(a1/1-r).
Convergent Sequence
When a sequence has a limit L that exists.
Two steps to find if a sequence is convergent.
1. Find a formula for the nth term, or an, of the sequence.
2. Find the limit of that formula as n approaches infinity. If the limit exists, the sequence is convergent.
If L does not exist, then the sequence is divergent.
Ratio Test
A way to find if a SERIES is convergent.
Three instructions:
Form a ratio of an+1 divided by an.
TAke the absolute value of this ratio.
Take the limit as n->infinity.
If L<1 the series converges
If L>1 the seires diverges
If L=1, then it is inclusive.
Steps for solving.
Step 1: Idneitfy the general term asubn
Step 2: Find an expression for asub(n+1)
Step 3: Build the ratio
Step 4: Simplify the ratio.
Step 5: Take hte absolute value
Step 6: Take the limit as n approaches infinity
Step 7: Comment on the result
Root Test
A simple test that tests for absolute convergence of a series, meaning the series definitely converges to some value. It will not tell you what it converges to.
Lim n->infinity abs|an|^(1/n)
If L<1 then the series absolute converges
If L>1, then the series diverges
If L=1 then the series could converge or diverge.
Integral Test
As long as the function that models the series is monotonic decreasing, you set up an improper integral for the function that models the series; if the improper integral diverges, then teh series diverges, and if the improper integral converges to a finite value, then the series converges.
The test will work if at some point, the terms of teh series become positive and decreasing.
Monotonic definition: A function or quantity varying in such a way that it either never decreases or never increases.
Integral test does not give you the sum of the series just whether it increases or decreases.
P Test
P>1 P-series converges
P<1 P-series diverges
Comparing converging and diverging series
If we know a series converges, then a series smaller then that one will also converge.
If we know a seires is divergent, then a series bigger then that one will also diverge.
Taylor Series
F(x)=fa+*f^(i)(x-a)^n)/n!
Where i=0 at the beginning.
Convergence definition
As we include more and more terms, the sum of terms is not growing without bounds.