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Solved problems on integral test and harmonic series.

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Presentation on theme: "Solved problems on integral test and harmonic series."— Presentation transcript:

1 Solved problems on integral test and harmonic series

2 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series integral test Let f be a positive decreasing function, and let a k = f(k).

3 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series Harmonic series The series is called the Harmonic Series. Using the Integral Test for the function we prove that the Harmonic Series diverges.

4 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series error estimate by the integral test Let f be a positive decreasing function, and let a k = f(k). If the series converges by the integral test, then Error of the approximation by the partial sum.

5 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series Determine whether the following series converge or diverge. OVERVIEW OF PROBLEMS 12 34 5

6 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series OVERVIEW OF PROBLEMS 6 7 8

7 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series Problem 1 INTEGRAL TEST Solution

8 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series INTEGRAL TEST Solution(contd)

9 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series Problem 2 INTEGRAL TEST Solution

10 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series INTEGRAL TEST Solution(contd)

11 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series INTEGRAL TEST Solution(contd)

12 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series Problem 3 INTEGRAL TEST Solution

13 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series INTEGRAL TEST Solution(contd)

14 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series Problem 4 Solution INTEGRAL TEST

15 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series INTEGRAL TEST Solution(contd)

16 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series Problem 5 Solution INTEGRAL TEST

17 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series INTEGRAL TEST Solution(contd)

18 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series Problem 6 INTEGRAL TEST

19 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series INTEGRAL TEST Solution

20 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series Problem 7 INTEGRAL TEST

21 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series INTEGRAL TEST Solution

22 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series INTEGRAL TEST In order to estimate the sum with error <0.001, we have to find out how many terms we need to take in our approximation. In other words, we need to find out M so that Solution(contd)

23 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series INTEGRAL TEST Solution(contd)

24 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series Problem 8 INTEGRAL TEST

25 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series INTEGRAL TEST Solution

26 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series INTEGRAL TEST Solution(contd)

27 Mika Seppälä: Solved Problems on Integral Test and Harmonic Series INTEGRAL TEST Solution(contd)

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