Less19HarmSeriesIntTestPSeries

Question 1

What is the harmonic series and what is unusual about it?

Answer 1

_n=1Σ^{∞ 1}/_n,

The terms tend to 0, but it diverges.

Because the series uses integers, the boxes are above the curve.

Question 2

What type of problem is the harmonic series?

Answer 2

an integral problem

Question 3

What is an improper integral?

Answer 3

An improper integral is a definite integral that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration

Question 4

Restate the Harmonic Series as an improper integral.

Answer 4

₁{^{∞ 1}/_x dx

Question 5

How do you restate improper integral problems?

Answer 5

Substitute b for ∞.

lim_b→∞ (₁{^{b 1}/_x dx),

lnb - ln1 = lnb - 0 = lnb

It diverges because lnb→∞ (very slowly)

Question 6

What does the integral test do?

Answer 6

It compares and improper integral and its series. Either both converge or both diverge.

Question 7

What are the requirements for the integral test?

Answer 7

The function must be positive, continuous and decreasing.

Question 8

How do you set up the integral test for _n=1Σ^∞1/_(n²+1)?

Answer 8

₁{^{∞ 1}/_(n²+1) dx

Question 9

What do you do next in the integral test?

Answer 9

lim_b→∞1{^b1/_(x²+1) dx

Question 10

What is the next step?

Answer 10

limb→∞[arctanx]1b

because the derivative of arctan is 1/(x²+1):

d/dx(arctanx) = 1/(x² + 1) or

d/dx(tan^-1) = 1/(x²+1)

Question 11

How do you solve it?

Answer 11

arctan∞ - arctan1 = π/2(90°) - π/4(45°) = π/4

It converges to π/4

Question 12

What does this mean about the original series?

Answer 12

The original series converges, but not necessarily to π/4

Question 13

Does _n=1Σ^∞1/_n³ converge?

Answer 13

f(x) = 1/x³

₁{^{∞ 1}/_x³ dx = x^-3

lim_b→∞[x^-2/-2]₁^b

0 - - ½ = ½

The improper integral coverges to ½

No one know what the series converges to.

Question 14

Does this series converge?

_n=1Σ^{∞ 1}/_√n

Answer 14

f(x) = x^-½

this is positive, continuous and decreasing

x^½/½ = lim_b→∞[2x^½]₁^b

^{this always gets larger, so diverges}

Question 15

What is a P-series?

Answer 15

1/n^p

Question 16

What is the P-series convergence test?

Answer 16

If P > 1, it converges.

If it is < 1 it diverges.

At 1, you have the harmonic series.

Question 17

What do _n=1Σ^∞1/_n², _n=1Σ^∞n³ and _n=1Σ^∞n⁴ converge to?

Answer 17

π²/6,

unknown,

π⁴/90

Question 18

What is the natural relationship between the harmonic series and logarithms?

Answer 18

lim_b→∞(₁{^{b 1}/_x dx),

-lim_b→∞[lnx]₁^b

Question 19

What is the Euler-Mascheroni Constant?

Answer 19

It is the difference between the partial sum and the natural log of each term in the harmonic series:

a₁=s1 - ln1

a₂=s₂ - ln2, etc.

The differences a₁, a₂, ,,,a_large → the constant ≈ .577 also called Gamma

Question 20

Does the starting term affect the divergence or convergence?

_n=5Σ^∞ e^-n

Answer 20

No. It’s the latter term that matters. It may affect what it converges to.

₅{^∞ e^-x dx

lim_b→∞ [-e^-b - - e^-5]

0 - e^-5 = e^-5

It converges to e^-5

Question 21

Does _n=1Σ^∞ 1/n^π converge?

Answer 21

P-series with π >1, so it converges.

Question 22

Does this converge?

_n=1Σ^{∞ n}/_√(n²+1)

Answer 22

This goes to n/n → 1 so they aren’t getting smaller.

It diverges.