Summary of Convergence Tests for Series and Solved Problems Integral Test Ratio Test Root Test Comparison Theorem for Series Alternating Series

Test Test Quantity Converges if Diverges if Ratio q < 1 q > 1 Root r < 1 r > 1 Integral Int < ∞ Int = ∞ The above test quantities can be used to study the convergence of the series S. In the Integral Test we assume that there is a decreasing non-negative function f such that ak = f(k) for all k. The Test Quantity of the Integral Test is the improper integral of this function. Mika Seppälä: Series

Comparison Test and the Alternating Series Test Assume that 0≤ ak ≤ bk for all k. If the series Alternating Series Test 1 2 Mika Seppälä: Series

Error Estimates Error Estimate by the Integral Test Error of the approximation by the Mth partial sum. Error Estimates by the Alternating Series Test This means that the error when estimating the sum of a converging alternating series is at most the absolute value of the first term left out. Mika Seppälä: Series

Overview of Problems 1 2 3 Mika Seppälä: Series

Overview of Problems 4 5 6 7 8 9 Do the above series 4-5 and 7 – 9 converge or diverge? 10 11 12 Mika Seppälä: Series

Overview of Problems 13 14 15 Do the series in 13 – 16 converge? 16 17 18 19 20 Do the series in 19 – 20 converge? Mika Seppälä: Series

Overview of Problems 21 22 24 25 23 26 27 Do the series given in Problems 23 – 29 converge? 28 29 30 Mika Seppälä: Series

Comparison Test 1 Solution Mika Seppälä: Series

Comparison Test 2 Solution Mika Seppälä: Series

The Comparison Test 3 Solution The series a) needs not converge. Example: Mika Seppälä: Series

The Comparison Test 3 Solution (cont’d) The series b) does converge. <1 Mika Seppälä: Series

The Integral Test 4 Solution Mika Seppälä: Series

Comparison Test From Applications of Differentiation. 5 Solution Mika Seppälä: Series

Partial Fraction Computation 6 Solution These terms cancel. Mika Seppälä: Series

Comparison Test 7 Solution Mika Seppälä: Series

The Integral and the Comparison Tests 8 Solution Mika Seppälä: Series

The Integral Test 9 Solution Hence the series diverges by the Integral Test. Mika Seppälä: Series

The Integral Test 10 Solution Computing 1000th partial sum by Maple we get the approximation 1.6439. The precise value of the above infinite sum is π2/6≈1.6449. Mika Seppälä: Series

The Comparison Test 11 Solution You can show this also directly by the Integral Test without referring to the Harmonic Series. Mika Seppälä: Series

The Comparison Test 12 Solution Mika Seppälä: Series

The Alternating Series Test 13 Solution Mika Seppälä: Series

The Alternating Series Test 14 Solution Mika Seppälä: Series

The Alternating Series Test 15 Solution Mika Seppälä: Series

The Alternating Series Test 16 Solution This follows from the fact that the sine function is increasing for 0≤x≤π/2. Mika Seppälä: Series

The Alternating Series Test 17 Solution Mika Seppälä: Series

The Alternating Series Test 18 Solution Mika Seppälä: Series

The Alternating Series Test 19 19 Solution Use l’Hospital’s Rule. Mika Seppälä: Series

The Alternating Series Test 20 Solution Use l’Hospital’s Rule. Mika Seppälä: Series

The Integral Test 21 Solution Mika Seppälä: Series

The Integral Test 22 Solution This requires that p≠1. If p=1, the corresponding improper integral diverges. Mika Seppälä: Series

The Root Test 23 Solution Use the Root Test. Mika Seppälä: Series

The Ratio Test 24 Solution Use the Ratio Test. Mika Seppälä: Series

The Comparison Test 25 Solution Use the Comparison Test. According to Problem 21. Conclude that the series converges. Mika Seppälä: Series

The Ratio Test 26 Solution Use the Ratio Test. Mika Seppälä: Series

The Ratio Test 27 Solution Use the Ratio Test. Mika Seppälä: Series

The Ratio Test 28 Solution Conclude that the series converges by the Ratio Test. Mika Seppälä: Series

The Ratio Test 29 Solution Observe that for all positive integers n, sin(n) + cos(n) ≠0. Hence, for every n, an≠ 0, and the above ratio is defined for all n. The series converges by the Ratio Test. Mika Seppälä: Series

The Ratio Test 30 Solution Use the Ratio Test. Mika Seppälä: Series