BC Calculus Unit 4 Day 5 Comparison Tests

Summary So Far Test for Divergence (DV or INCONCLUSIVE) –Limit of terms compared to zero Geometric Series (DV or CV with CV Value) –r value compared to 1 P-Series (Including Harmonic Series) –p value compared to 1 Telescoping Test (DV or CV with CV Value) Integral Test (DV or CV without CV Value)

Today’s Tests... Direct Comparison (D.C.T.) Limit Comparison (L.C.T.) Alternating Series

Determine whether the following is convergent or divergent Before we look at the D.C.T Divergent by Integral Test OR by “comparison” to the Harmonic Series

The comparison to Harmonic works like this... Since, And because diverges then diverges too.

For the next example, we will first need to determine whether converges or diverges. Convergent to By Geometric Series Test,

Compare the following two series

Conclusion... Since, And because converges to 1 then converges to some value less than 1.

Direct Comparison Test (D.C.T.)

Picking a Comparison Pick a series that is known to converge or diverge based on Geometric Test or P-Series Test Make sure you can confirm the appropriate inequality Consider only leading terms of the numerator and denominator when picking

Examples (Practice) Use the D.C.T to determine convergence/divergence of the following series:

Look for a SIMPLE series to compare to If the series is The picked comparison would likely be,

Limit Comparison Test (L.C.T)

1.Pick the comparison (“known”) series: 2. 3.By the Limit Comparison Test will also converge Example Problem Determine the convergence/divergence of : Converges by P-Series Test p=3>1

Examples (Practice) Use the L.C.T. to determine convergence/divergence of the following series:

Special cases of the Limit Comparison Test (L.C.T)

Summary So Far Test for Divergence (DV or INCONCLUSIVE) –Limit of terms compared to zero Geometric Series (DV or CV with CV Value) –r value compared to 1 P-Series (Including Harmonic Series) –p value compared to 1 Telescoping Test (DV or CV with CV Value) Integral Test (DV or CV without CV Value) D.C.T. (Both DV or Both CV) –Must satisfy inequality L.C.T. (Both DV or Both CV) –Evaluate limit of ratio and compare to 0 or infinity