Infinite Series

Question 0

What is a series?

Answer 0

A series is a sum.
Example: 1/2 + 1/4 + 1/8 + 1/16 +…
Tip: has addition signs.

Question 1

What is a sequence?

Answer 1

A sequence is a function.
Example: 1/2, 1/4, 1/8, 1/16,…
Tip: a sequence has commas.

Question 2

When does a sequence converge?

Answer 2

For any sequence {a sub n}, if the limit as n approaches infinity equals L, where L is a real number, then the sequence CONVERGES to L.

Question 3

When does a sequence diverge?

Answer 3

For any sequence {a sub n} , if the limit as n approaches infinity is NOT a real number, then the sequence DIVERGES.

Question 4

What does diverging sequences do?

Answer 4

• Increase forever(the limit of a sub n equals infinity).
• Oscillate like a sequence 1,-1,1,-1…
• Exhibit no pattern at all- this is rare.

Question 5

Why do a sequence and it’s related function go hand in hand?

Answer 5

If the limit of the function as x approaches infinity is some finite number, L, then the limit of the sequence is also L, and thus, the sequence converges to L. Also the graph of the convergent function/sequence pair has a horizontal asymptote at L.

Question 6

How do you determine limits of sequences with L’Hopital’s rule?

Answer 6

To use L’Hopital’s rule, take the derivative of the numerator and the derivative of the denominator.
Tip: you may need to apply it twice.

Question 7

What is an infinite series?

Answer 7

It is simply the adding up of the infinite number of terms of a sequence .

Question 8

How do you find an infinite sum?

Answer 8

You take the limit- just like you do for improper(infinite) integrals.

Question 9

What is a Partial Sum?

Answer 9

The nth partial sum, S sub n, of an infinite series is the sum of the first n terms often series.
Example: S sub 1 =1/2, S sub 2=1/2 + 1/4 = 3/4, S sub 3 =1/2 + 1/4 + 1/8=7/8…. S sub n=1/2+1/4+1/8+1/16+1/32+1/64+…+1/2^n

Question 10

What is the main point of infinite series?

Answer 10

To figure out whether such a sequence of partial sums CONVERGES- hones in on a finite number- or DIVERGES. If the sequence of partial sums converges, you say the series converges;otherwise, the sequence of partial sums diverges and you say that the series diverges.

Question 11

How do you find the limit of a sequence of partial sums?

Answer 11

You find the nth partial sum, S sub n, then take the limit.

Question 12

What is the nth Term Test?

Answer 12

If the limit of a sub n, as n approaches infinity, does not equal zero, then the sum of a sub n diverges.
As to say: if the terms of the series’ underlying sequence do not converge to zero, then the series must diverge.
WARNING when the terms of a series converge to zero, that does NOY guarantee that the series converges.

Question 13

What is a Geometric series and what is it’s form?

Answer 13

A geometric series is a series of the form: a+ar+ar^2+ar^3+ar^4+…= the sum of ar^n , with n=0 going to infinity.
a is called the leading term. Each term after the first equals the preceding term multiplied by r, which is called the ratio.
Example: 5+53+53^2+53^3…=5+15+45+135

Question 14

What is the geometric series rule?

Answer 14

If 0=1, the series diverges.

*NOTE: this rule works when -1<0 in which case you get an alternating series.

Question 15

What is the form of a p-series?

Answer 15

A p-series is of the form: the sum of 1/n^p to infinity, where n=1, equals 1/1^p + 1/2^p + 1/3^p + 1/4^p +… (Where p is a positive power) the p-series for p=1 is called the harmonic series.

Question 16

What is the p-series rule?

Answer 16

The p-series sum 1/n^p converges if p > 1 and diverges if p <= 1

Question 17

In short, what is direct comparison test?

Answer 17

If you have a series that’s smaller than a convergent benchmark series, then your series must also converge. And if your series is later than a divergent benchmark series, then your series must also diverge.

Question 18

What is the direct comparison test?

Answer 18

Let 0 <= b sub n for all n.

• If the sum of b sub n converges, then the sum of a sub n converges.
• if the sum of a sub n diverges, then the sum of b sub n diverges.