CALC BC Chapter 9 Theorems

Sequence

the list of numbers

Series

the sum of numbers

Geometric Sequence behaviors

As the limit of n → infinity

if |r| < 1, then the limit approaches to 0

if r = 1, then the limit approaches to 1

if |r|>1, then the limit is DNE

Therefore, the convergence is -1< r <= 1

Monotonic

r>0

Oscillate

r<0

Geometric ratio formula

a n+1 / an

Sum of Geometric Series

a (1 - r^n / 1 - r)

If the geometric series convergences, then what is the formula used?

a / 1-r

If |r| <1 in an infinite geometric series, then it converges

a / 1 - r

If |r| => 1 in a infinite geometric series,

then the series diverges

Divergence Test

If the series converges, then the limit is 0

If the limit is not 0, then the series diverges

Always, sometimes, or never

If the limit is 0, the series converges

Sometimes

Harmonic Series

the limit as n → infinity of 1/n = 0, but the series itself diverges

P series

if p >1, then the series converges

if p <= 1, then the series diverges

Conditions of the integral test

f must be continuous, positive and decreasing

What does the integral test tell you?

-both series and integral can either both converge OR diverge

-in converge, the value of the integral DOES NOT equal to the value of the series

What is the behavior of the ratio test?

• if 0 <= r < 1, then the series converges

• if r = 1, then the series is inconclusive and needs further testing

• if r >1, then the series diverges

What is the ratio test formula?

What are the conditions for the ratio test?

When should you use it?

• r = lim k→ infinity (a k+1 /a k)

• An infinity series with positive terms

  • Used for factorials, exponentials or “n” powers

What is the behavior of the root test?

• if 0<= p < 1, then the series converges

• if p = 1, then the series is inconclusive and needs further testing

  • if p > 1, then the series diverges

What is the formula for the root test?

What are the conditions for the root test?

• formula = the limit as k → infinitity of the root of k

  • conditions = an infinite series

Direct comparison statements:

• if the big series (b) converges, then the smaller series (a) converges, too

• if the big series (a) diverges, then the smaller series (b) diverges, too

• if the smaller series diverges, the bigger series MAY diverge OR converge

Alternating Series conditions:

• a series with (-1)^k or (-1)^k+1

• Can converge if

  • the series is nonincreasing in magnitude

  • the limit k→infinity = 0 (divergence test)

Alternating harmonic series will converge or diverge?

converge

• the limit = 0 AND the sum grows smaller to approach 0 and the terms grow smaller

What is the behavior of the remainder in an alternating series?

|R| <= a (n+1)

Absolutely converging:

Absolute value of series is converging, therefore the alternating series is converging

Conditionally converging:

Absolute value of series if diverging, but the alternating series is converging

Divergent

Both the absolute value and alternative series if diverging

When is geometric series convergent?

|r|<1