Sequences And Series Flashcards

1
Q

Arithmetic series sum

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Finding the nth term of a geometric sequence

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

Geometric Sequence

A

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)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

Geometric Series

A

Adding up a geometric sequence.

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

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

A

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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

Geometric series where n is to infinity.

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Geometric series for an infinite series

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

Convergent Sequence

A

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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Ratio Test

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Root Test

A

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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Integral Test

A

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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

P Test

A

P>1 P-series converges

P<1 P-series diverges

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

Comparing converging and diverging series

A

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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

Taylor Series

A

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

Where i=0 at the beginning.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

Convergence definition

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

MacLauran Series

A

A Taylor series evaluated at a=0.

17
Q

Taylor Polynomial

A

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

18
Q

Making a sequence a continuous function with L’Hopitals rule

A

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.

19
Q

Overlapping limits of functions adn limits of series

A

Check page 621.
Squeeze theorem also applies.

20
Q

Monotone sequence

A

Whether is a sequence is increasing or decreasing.

21
Q

Bounded Sequence

A

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.

22
Q

Nth term test

A

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.

23
Q

Harmonic series

A

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

24
Q

Absolute convergence vs conditional convergence

A

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.

25
Q

P Series or Hyperharmonic series

A

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.

26
Q

Limit Comparison Test

A

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.