Flashcards in Chapter 11 Review (Test 1) Deck (16):

1

## Sequence

###
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.

2

## Arithmetic sequence

###
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

3

## Geometric Sequence

###
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.

4

## limits of sequences

###
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

5

## A sequence converges to L if and only if:

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

6

## Recursive sequences

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

7

## Limits of Recursive Sequences

###
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.

8

## Series

### Adding terms

9

## Infinite series

### a series with an infinite number of terms.

10

## Convergence of an infinite series

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

11

## Convergence of an Infinite Geometric Series

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

12

## Divergence of an Infinite Geometric Series

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

13

## Telescoping Series

### Ex: 1 divided by 1*2 plus 1 divided by 2*3 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.

14

## Harmonic Series

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

15

## Divergence Test

###
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).

16