Calculus BC Unit 4 Day 3 Test for Divergence Integral Test P-Series (Including Harmonic)

Test for Divergence Theorem (a.k.a The n th Term Test) If Does Not Exist OR if Then the series is divergent

Test for Divergence

CAUTION You can never conclude CONVERGENCE using the Test for Divergence. In other words, does not necessarily mean is convergent. means the Test for Divergence is INCONCLUSIVE.

Summary So Far Geometric Series – Examine r value Test for Divergence – Divergent OR Inconclusive Next.... Integral Test

The Integral Test Suppose f is a continuous, positive, decreasing function for all and Then:

Let Is f(x) continuous? ____ Positive? ___ Decreasing? ____ (How could you use derivatives to prove your answer?) Exploring the following P-series using the integral test:

Now we can do the Integral Test It does NOT mean that it converges to 1. Just that it converges.

Integral Test Example: Is f(x) continuous? ____ Positive? ___ Decreasing? ____

Could we have used the Test for Divergence? n th Term Test (a.ka. Test for Divergence):

Integral Test Example #2: Is f(x) continuous? ____ Positive? ___ Decreasing? ____ How do you know?

Integral Test Example: Is f(x) continuous? ____ Positive? ___ Decreasing? ____ How do you know?

Exploring the Harmonic Series The name “Harmonic series” comes from the world of music and overtones, or harmonics. The wavelengths of the overtones of a vibrating string are Source: Wikipedia.com The name “Harmonic series” comes from the world of music and overtones, or harmonics. The wavelengths of the overtones of a vibrating string are Source: Wikipedia.com

The Harmonic Series is an example that confirms the converse of the following theorem to be false! If the series is convergent, then The CONVERSE is NOT TRUE

is convergent IF and divergent IF The Harmonic Series is an example of a P-Series A P-Series is of the form P = 1 gives the harmonic series

Determine convergence or divergence ConvergeDiverge

Summary So Far Geometric Series (this is the only series we can find a converging value so far!) – Examine r value Test for Divergence – Divergent OR INCONCLUSIVE Integral Test P-Series (Including Harmonic Series)

Remember.... If the sequence of the partial sums converges to some value S ( ) then the series converges AND we state that

Facts about Telescoping Series “Evaluate” means find the sum, which will only exist for convergent series. All telescoping series are convergent. Telescoping series and geometric series are the only two types of series that we can find the “sum” for.

“Evaluate” the following Telescoping Series First, “decompose” using partial fractions: Second, separate the two sums, and expand to k. Fourth, Find the limit at k-> ∞: Five, finalize your answer: Third, subtract the two sums:

“Evaluate” the following Telescoping Series

Example #3—YOU TRY

The k th partial sum is And So the series Converges to 1 Example #2—SOLUTION

Telescoping Test…Ex. P. 3 #19 Let’s look at partial sums:

The kth partial sum is And So the series Converges to 5/6 Telescoping Test…Ex. P. 3 #19