Calculus II Exam 3

Identify the P-Series

n^# or sqrtn

Identify the Geometric Series

r is directly to power of n/k

a is what’s not attached

#ⁿ

Solving P-Series

1. Identify P aka the power that n/k is to

   • for multiple of them, P is the sum

2. Compare P to 1

   • Absolutely Converges if P is greater than 1

   • Diverges if P is less than or equal to 1

Solving Geometric Series

1. Identify r aka the # that is to the power of k/n

   • if in denominator, only take the value of what is directly attached to k/n, but remember if its a fraction or not being in the denominator

   • if # is to like 2n, keep the power of 2 in r

2. Ask: Is |r| < 1?

   • Yes → Absolutely Converges to s= a/1-r

     • to find a, plug in number n is = to in sigma

   • No → Diverges

To get rid of square roots…

Multiply by the conjugate and use (A-B) (A+B) = A²-B²

Identify Telescoping Series

Terms subtract and when expanded, cancel out only leaving first and last term

Solving Telescoping Series

1. Expand the series

   • last term is just the second value from the original series but with n’s

2. Cancel out terms except first and last which will be combined

3. Take the limit of this

   • this is the sequence of partial sums

4. If it converges, then so does the series

5. If it diverges, then so does the series

Identifying Almost P-Series

if k^# or sqrtk is added/subtracted by another # or variable (s)

Identifying Almost Geometric Series

if #^k is added/subtracted by another # or variable

Solving Almost Series

1. Determine if almost p-series or geo series

2. Take away lower degree terms to make look alike series Σ bn

3. State “Compare to [insert Σ bn]

4. Decide if using Limit Comparison Test or Direct Comparison Test

5. State the test

6. Perform the test

7. Clearly state conclusion

Limit Comparison Test

Use when taking away more than one term from numerator and denominator to create Σ bn

1. Find if Σ bn converges or diverges

2. Set up Lim n→ ∞ an/bn = L

3. Solve for L and determine if it is positive and finite

   • if so, then series will do what Σ bn does

4. If L is…

   • L = 0 or pos finite # and Σ bn converges, then og series converges OR tog series will follow what Σ bn does

   • L = ∞ and Σ bn diverges, og series diverges too

Direct Comparison Test

Use if only taking one term away to make Σ bn

1. Find if Σ bn converges or diverges

2. Compare the size of Σ an and Σ bn

   • Smaller than convergent is convergent

     • an smaller than bn and bn is convergent = convergent

   • Bigger than divergent is divergent

     • an bigger than bn and bn is divergent = divergent

Solving Series Not P, Geo, Q-log, Telescoping but Positive Term Series

1. Try Divergence Test (see if lim n→ an goes to 0

   • doesn’t go to 0 = diverges, you are done

   • does go to 0 = use Integral Test

2. Try Integral Test

3. Check the following conditions to use Integral Test

   a) When plugging in # for k into denominator, can’t equal zero

   B) Derivative of f(x) must be negative

4. Compute ∫ ∞ 1 f(x)

5. Determine if integral converges or diverges

   • series will do what integral does

Solving Series w/ All or Multiple Terms to a Single n Power

Use Root Test

1. Set up Lim n → ∞ ⁿsqrt( | an | ) = L

2. Cancel terms if applicable

3. Determine value of L

   • Converges if L less than 1

   • Diverges if L greater than 1

   • Try something else if L =1

Solving Series with Factorials or Terms to Power of n

Use Ratio Test

1. Set up Lim n→ ∞ a_n+1/an = L

2. Cance terms if applicable

3. Determine value of L

   • Converges if L is less than 1

   • Diverges if L is greater than 1

   • Try something else if L = 1

How to Write an+1

Add 1 to each n/k term

n^# → (n+1)^#

eⁿ or #ⁿ → _ⁿ⁺¹

n! → (n+1) !

ke^-k (k+1)(e^-(k+1))

nⁿ → (n+1)ⁿ⁺¹

Constant #s don’t apply and stay the same

Identify Q-Log Series

P-series with log in numerator

Solving Q-Log Series

1. Identify q aka the power k/n is to

2. Compare q to 1

   • Converges if q is greater than 1

   • Diverges if q is less than or equal to 1

Identify Alternating Series

When expanded, signs alternate

• (a₁, -a₂, a₃, -a₄…)

Solving Alternating Series

Use Alternating Series Test

1. Check two conditions; if both happen, converges, if not, diverges

   a) Does Lim n→ ∞ an = 0?

   b) Is a_n+1 less than an?

   • more on bottom makes term smaller

Determining Absolute Cvt, Conditional Cvt, or Divergence

1. Check for absolute convergence using Ratio Test/Root Test w/ absolute bars, Comparison Test, or Integral Test

2. For Ratio/Root tests, compare L to 1

   • Absolute convergent is L is less than 1

   • NOT absolute convergent if L is greater than 1

3. If not Absolute Cvt, check Conditional Cvt using Alternating Series Test

4. Use Alternating Series Test

   • if all conditions are met, converges, if not, diverges

Identify Alternate P-Series

They ALL converge

Finding Interval of Convergence

Use Ratio Test or Root Test w/ absolute bars both < 1

1. Decide which test to use

2. Plug in and cancel terms

3. Take out |x| or |x-#| or |x-#| / # and put in front of Lim

4. Solve for limit of an and multiply it by x

5. Then solve for x; what’s on the other side of < is R

6. Identify the center and draw a number line with center as middle and one point R distance to the left and another R distance to the right

7. Test each endpoint to see which gets brackets and parentheses

• 1. Let x = #

• 2. Plug in # into x in original equation

• 3. If diverges → don’t include. If converse → include

Taylor Series Centered at x = c

Taylor Series Centered at c = 0

Solving for Taylor Series

1. Find the first five derivatives of f(x)

2. Then, plug in value for center into x in the results for the derivatives

3. Plug into formula, simplify or cancel factorials if needed

4. Look and identify the series, check by plugging in to make sure

Maclaurin Series for e^x

Maclaurin Series for sinx

Maclaurin Series for cosx

Maclaurin Series for 1 / 1 - x

Maclaurin Series for ln(1 + x)

Maclaurin Series for arctanx

Maclaurin Series for ( 1 + x ) ^9

Finding Absolute/Conditional Cvt or Divergence for P-Series/Geo Series

Use the regular way to find convergence for them

• If converges, is absolutely convergent

• If diverges, it diverges and you’re done

Finding Absolute/Conditional Cvt or Divergence for Almost Series

Use Comparison Tests to Test for Absolute Convergence

• if converges, is absolutely convergent

• if diverges, check for conditional convergence

Finding Absolute/Conditional Cvt or Divergence using Integral Test

1. Check for absolute convergence using Integral Test

   • If converges, then absolutely converges

   • If diverges, check for conditional convergence

2. If needed, use AST to check for conditional convergence