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

2. 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

3. Comparison Test and the Alternating Series Test
Assume that 0≤ ak ≤ bk for all k. If the series
Alternating Series Test 1 2
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4. 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.
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5. Overview of Problems 1 2 3
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6. Overview of Problems 4 5 6 7 8 9
Do the above series 4-5 and 7 – 9 converge or diverge? 10 11 12
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7. Overview of Problems 13 14 15 Do the series in 13 – 16 converge? 16 1718 19 20
Do the series in 19 – 20 converge?
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8. Overview of Problems 21 22 24 25 23 26 27
Do the series given in Problems 23 – 29 converge? 28 29 30
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9. Comparison Test 1
Solution
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10. Comparison Test 2
Solution
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11. The Comparison Test 3 Solution The series a) needs not converge.
Example:
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12. The Comparison Test 3 Solution (cont’d) The series b) does converge.
<1
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13. The Integral Test 4
Solution
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14. Comparison Test From Applications of Differentiation. 5 Solution
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15. Partial Fraction Computation6
Solution
These terms cancel.
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16. Comparison Test 7
Solution
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17. The Integral and the Comparison Tests8
Solution
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18. The Integral Test 9 Solution
Hence the series diverges by the Integral Test.
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19. The Integral Test 10 Solution
Computing 1000th partial sum by Maple we get the approximation The precise value of the above infinite sum is π2/6≈
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20. The Comparison Test 11 Solution
You can show this also directly by the Integral Test without referring to the Harmonic Series.
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21. The Comparison Test 12
Solution
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22. The Alternating Series Test13
Solution
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23. The Alternating Series Test14
Solution
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24. The Alternating Series Test15
Solution
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25. The Alternating Series Test16
Solution
This follows from the fact that the sine function is increasing for 0≤x≤π/2.
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26. The Alternating Series Test17
Solution
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27. The Alternating Series Test18
Solution
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28. The Alternating Series Test19 19
Solution
Use l’Hospital’s Rule.
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29. The Alternating Series Test20
Solution
Use l’Hospital’s Rule.
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30. The Integral Test 21
Solution
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31. The Integral Test 22 Solution
This requires that p≠1. If p=1, the corresponding improper integral diverges.
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32. The Root Test 23
Solution
Use the Root Test.
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33. The Ratio Test 24
Solution
Use the Ratio Test.
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34. The Comparison Test 25 Solution Use the Comparison Test.
According to Problem 21.
Conclude that the series converges.
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35. The Ratio Test 26
Solution
Use the Ratio Test.
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36. The Ratio Test 27
Solution
Use the Ratio Test.
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37. The Ratio Test 28 Solution
Conclude that the series converges by the Ratio Test.
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38. 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.
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39. The Ratio Test 30
Solution
Use the Ratio Test.
Mika Seppälä: Series