Sequences And Series

Question 1

Arithmetic series sum

Answer 1

n/2(2a+(n-1)d))

Question 2

Finding the nth term of a geometric sequence

Question 3

Geometric Sequence

Answer 3

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)

Question 4

Geometric Series

Answer 4

Adding up a geometric sequence.

S=a1(1-r^n)/(1-r). This is the sum of any finite geometric series.

Question 5

Geometric series sum for infinite series if r is between 0 and 1

Answer 5

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.

Question 6

Geometric series where n is to infinity.

Answer 6

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

Question 7

Geometric series for an infinite series

Answer 7

If the r is less then 1, then the formula turns into series=(a1/1-r).

Question 8

Convergent Sequence

Answer 8

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.

Question 9

Ratio Test

Answer 9

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

Question 10

Root Test

Answer 10

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.

Question 11

Integral Test

Answer 11

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.

Question 12

P Test

Answer 12

P>1 P-series converges

P<1 P-series diverges

Question 13

Comparing converging and diverging series

Answer 13

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.

Question 14

Taylor Series

Answer 14

F(x)=fa+*f^(i)(x-a)^n)/n!

Where i=0 at the beginning.

Question 15

Convergence definition

Answer 15

As we include more and more terms, the sum of terms is not growing without bounds.

Question 16

MacLauran Series

Answer 16

A Taylor series evaluated at a=0.

Question 17

Taylor Polynomial

Answer 17

The Polynomial we get by keeping some but not all of the terms.

Question 18

Making a sequence a continuous function with L’Hopitals rule

Answer 18

If we can form a continuous function for a series f(x) where x is real and f(n)=an

We can say limx->infinity f(x)=L. Then limn->infinityan=L. This means that the answer we get when we use l-hospitals rule is the actual value that the sequence converges to.

Question 19

Overlapping limits of functions adn limits of series

Answer 19

Check page 621.
Squeeze theorem also applies.

Question 20

Monotone sequence

Answer 20

Whether is a sequence is increasing or decreasing.

Question 21

Bounded Sequence

Answer 21

A sequence that for all values of n will never get above a certain number or never get below a certain number. A bounded Monotone sequence will converge.

Question 22

Nth term test

Answer 22

If limn->infinityan does not equal zero. Then the sum of an diverges.

If limn->infinityan=0 then further investigation is necessary to see if it converges or diverges.

Question 23

Harmonic series

Answer 23

This means the series diverges even though the limn->infinity an=0.

Question 24

Absolute convergence vs conditional convergence

Answer 24

AN infinite series converges absolutely if the absolute value of an as it approaches infinity also converges.

IF the absolute value of hte series does not converge then it is conditionally convergent.

Question 25

P Series or Hyperharmonic series

Answer 25

P series is sum 1/n^p where p is a positive real number. If p>1, converges, if p< or equal to 1, diverges.

Question 26

Limit Comparison Test

Answer 26

If an>0 and Ben>0

If the limits->infinityabs|An/Bn| and is finite and non-zero then teh series either both converge (absolutely) or diverge.