Calculus2 Lesson19&20

Question 1

_n=1Σ^∞ 1/n = 1 + ½ + ⅓ + ¼ + . . .

Answer 1

Harmonic series

Question 2

What is an improper integral?

Answer 2

a definite integral with one or both endpoints undefined or infinite

Question 3

How do you deal with an improper integral?

Answer 3

You treat it as a limit from 1 to b with b → ∞ (or the undefined value)

Question 4

What must a series and its improper integral have in common?

Answer 4

They both either converge or diverge, but not necessarily to the same number.

Question 5

Does the harmonic series converge or diverge?

Answer 5

It diverges.

Question 6

What is the integral test?

Answer 6

It compares the infinite series with its improper integral of its function. The function must be positive, continuous and decreasing in the range.

Question 7

What is the integral of 1/(x² + 1) dx?

Answer 7

arctanx

Question 8

What is the derivative of arctanx ?

Answer 8

1/(x² + 1)

Question 9

What is the mystery of the series:

_n=1Σ^∞ 1/n³ ?

Answer 9

It converges but no one has determined to what number it converges.

Question 10

Does _n=1Σ^∞ 1/√n diverge or converge?

Answer 10

It diverges.

Question 11

What is a P-series?

Answer 11

1/n^P If P>1, it converges.

If P is between 0 and 1 (including 1) it diverges.

Question 12

How are the P-series and the Harmonic Series related?

Answer 12

The harmonic series is the boundary point. P = 1

Question 13

What is true if the terms of a series do not go to 0?

Answer 13

It diverges.

Question 14

What happens if the terms of a series do tend to 0?

Answer 14

It’s unknown. It could converge or diverge.

Question 15

Summary of P-Series from 1/n to 1/n⁴

Answer 15

1/n diverges

1/n² converges to π²/6

1/n³ converges to unknown point

1/n⁴ converges to π⁴/90

Question 16

What is the integral of 1/x?

Answer 16

lnx. This is the natural relationship between the harmonic series and logarithms.

Question 17

What is the partial sum from 1 to 100 in the harmonic series?

Answer 17

menu → calculus → #5 Sum

≈ 5.18738

Question 18

What is the Euler-Mascheroni constant?

Answer 18

S_n = _k=1Σⁿ1/k - lnn

In the harmonic series, as n gets larger, the difference between the partial sums and the lnn approach a finite limit around .57721

Question 19

What does the Euler-Mascheroni constant show?

Answer 19

Even though the harmonic series and the ln function both diverge, the difference between them converges.

Question 20

What does e^-5 converge to?

Answer 20

e^-5

Question 21

Does _n=1Σ^∞ 1/n^π converge or diverge?

Answer 21

P-series. P = π > 1, so converges