Chapter 11 Review (Test 1)

Question 1

Sequence

Answer 1

bracket a sub n bracket from n = 1 to infinity.

A sequence is a listing of values of a sub n as n goes from 1 to infinity.

Question 2

Arithmetic sequence

Answer 2

Each term is the sum of the previous term and a constant (called the common difference).
a sub n = a sub 1 plus (n-1)*d

Question 3

Geometric Sequence

Answer 3

each term is the product of the previous term and a constant (coalled the common ratio).
a sub n = a sub 1 * r to the n-1 power.

Question 4

limits of sequences

Answer 4

the sequence of {a sub n} converges to L if limit as n approaches infinity of a sub n = L.
the sequence of {a sub n} diverges if limit of a sub n diverges. (equals infinity)
*remember to test for even and odd terms when dealing with sequences with (-1) to the n or n+1 power

Question 5

A sequence converges to L if and only if:

Answer 5

the even numbered terms converge to L and the odd numbered terms converge to L.

Question 6

Recursive sequences

Answer 6

a sequence in which you are given an initial term and a formula to find the subsequent terms.

Question 7

Limits of Recursive Sequences

Answer 7

a sequence defined recursively will converge if:
limit as n approaches infinity of a sub n+1 = limit as n approaches infinity of a sub n = L.
Steps:
1. Set limit as n approaches infinity of a sub n equal to L.
2. Set limit as n approaches infinity of a sub n + 1 (with L substituted in) equal to L.

Question 8

Series

Answer 8

Adding terms

Question 9

Infinite series

Answer 9

a series with an infinite number of terms.

Question 10

Convergence of an infinite series

Answer 10

If an infinite series converges to L, then the sum equals L. An infinite series converges if the sequence of partial sums converge.

Question 11

Convergence of an Infinite Geometric Series

Answer 11

An infinite geometric series converges if absolute value of r < 1 AND it converges TO a sub 1 divided by 1 minus r/

Question 12

Divergence of an Infinite Geometric Series

Answer 12

An infinite geometric series diverges if the absolute value of r is greater than or equal to 1.

Question 13

Telescoping Series

Answer 13

Ex: 1 divided by 12 plus 1 divided by 23 plus 1 divided by 3*4 and so on. (not geometric). Turn into partial fractions and then plug k back in and cancel everything out.

Question 14

Harmonic Series

Answer 14

ex: sigma from k=1 to infinity of 1 divided by k. Diverges.

Question 15

Divergence Test

Answer 15

Let a sub k be the general term of sigma a sub k.
If limit as k approaches infinity of a sub k does not equal 0, then sigma a sub k diverges.
If limit as k approaches infinity of a sub k = 0, then sigma of a sub k may converge or diverge (test is inconclusive).

Question 16

Properties of Series

Answer 16

sigma of U sub k plus or minus v sub k = sigma u sub k plus sigma of v sub k.
sigma of c times u sub k equals c times sigma u sub k.