Cal 2 Midterm #2 Questions

Question 1

Explain the solution to this problem

Answer 1

All we must do is take the limit of each and ensure we choose the only an answer that doesnt approach 0.

a = 0
b = 1/(π/2)
c = 0
d = 0

Question 2

Explain the convergence of this limit

Answer 2

Since we see that there is a k¹’¹ in the denominator, we can choose a larger function to test and perform the limit comparison test. This will be bₖ = 1/k¹’⁰⁵ since this series converges

Now we can use our function:

limit k → ∞ aₖ/bₖ

this limit will lead to 1/∞ = 0 which is too convergent

This makes our series Absolutely Convergent

Question 3

Which of the choices match the McLaurin Series

Answer 3

we can quickly take away (d) cos(-x) and (e) because these two do not resemble the problem at all.

e⁻²ˣ-1 seems closest to the choice we want as we see an increase in the value in the numerator:

f(x) = e⁻²ˣ-1 ; f(0) = 0
f’(x) = -2e⁻²ˣ ; f’(0) = -2
f’‘(x) = 4e⁻²ˣ ; f’‘(0) = 4
f’’‘(x) = -8e⁻²ˣ ; f’’‘(0) = -8

f(x) = 1 - 2x + 4x²/2! - 8x³/3!

Question 4

How do we find the sum of this series ?

Answer 4

We can start with identifying this series as geometric since since a sum can be determined and there is a rate at which the swing’s height converges.

Σ10(5/7)ᵏ

Then use the geometric sequence equation: a/(1-r)

10/[1 - 5/7] = 35

Question 5

How to solve this

Answer 5

We can use the divergence test to solve this since:

limit as k → ∞ tan⁻¹(k) is just π/2 and this π/2 ≠ 0

This series Diverges

Question 6

Explain the solution

Answer 6

We can start by acknowlegeing that this is an alternating series since sinx oscilates between -1 and 1.

Since this is alternating, all we need to do is take the limit as k → ∞ of 1/k² to see if this is series is convergent. We should also insure that the absolute value of sinx is considered as the negativity of this value is not of or concern once we determine that is it is divergent or convergent

b

Question 7

Answer 7

Since we know that cosx oscilates between -1 and 1, we can write this as (-1)ᵏ.

since we know that 1/k² converges we can tell that this series Converges Absolutely by alternating test.

e

Question 8

What are the intervals of the Ratio test ?

Answer 8

limit as k → ∞ of |aₖ₊₁/aₖ| must be less than 1 to be convergent. if it is equal to 1 then we must use a different test

Question 9

What are the intervals of the Root test ?

Answer 9

limit as k → ∞ of kth root of |aₖ| must be less than 1 to be convergent. if it is equal to 1 then we must use a different test

Question 10

Answer this

Answer 10

We can see that this has the apperrance of a geometric series and can be broken down to match it:

Σ (-1)ᵏ(x²)ᵏ(x²)

We can now treat x as a constant and move the x² outside:

x² Σ (-x²)ᵏ; we can now use the geometric series formula to tell the function:

x²/[1 - (-x²)] → x²/[1+x²]