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W07 - HW Solutions

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02 - Squeeze theorem

Determine whether the sequence converges, and if it does find its limit:

  • (a)
  • (b)

Solution

(a)

Note that since ,

Divide all terms by .

Use squeeze theorem.

Thus,

We conclude that converges.

(b)

Note that .

Raise all terms to the power of

Use squeeze theorem.

Thus,

We conclude that converges.

07 - Limits and convergence

Determine whether each sequence converges or diverges (C or D). If it converges, find the limit. [Refer to HW for the problem statements.]

Solution

**( Find the limit. Note that the degree of the numerator and denominator is the same.

We conclude that converges.

(b)

Find the limit.

Because of the term, the limit does not exist.

We conclude that diverges.

(c)

Find the lim

We conclude that converges.

(d)

Find the limit.

We conclude that converges.

(e)

Find the limit.

We conclude that converges.

(f)

Find the limit.

We conclude that diverges.

(g)

Find the limit.

We conclude that diverges.

(h)

Find the limit. Note that factorial expressions increase at a faster rate than exponential expressions.

We conclude that diverges.

(i)

Find the limit.

We conclude that converges.

(j)

Find the limit.

We conclude that converges.

(k)

Find the limit.

We conclude that diverges.

(l)

Find the limit.

We conclude that converges.

(m)

Find the limit.

We conclude that converges.

(n)

Find the limit. Note that .

We conclude that diverges.

(o)

Find the limit.

We conclude that converges.

(p)

Find the limit.

We conclude that converges.

(r)

Find the limit.

We conclude that converges.

(s)

Find the limit.

We conclude that diverges.

(t)

Find the limit. Use squeeze theorem.

We conclude that converges.

09 - Series from its partial sums

Suppose we know that the partial sums of a series are given by the formula .

  • (a) Compute .
  • (b) Find a formula for the general term .
  • (c) Find the sum .

Solution

(a) Start by computing .

Compute by noting .

Compute by noting .

(b)

Note that in part (a), we used for all .

Note that the above formula is undefined at .

(c)

Find S by computing .

10 - Geometric series - partial sums and total sum

Consider the series:

  • (a) Compute a formula for the partial sum . (You may apply the known formula or derive it again in this case using the “shift method.“)
  • (b) By taking the limit of this formula as , find the value of the series.
  • (c) Find the same value of the series by computing and and plugging into

Solution

(a) Compute the first few terms of the sequence.

Evidently, this is a geometric sequence where , and .

Write out formula for for a geometric series.

Plug in relevant terms.

(b)

Find the limit.

(c) Use and .

11 - Total area of infinitely many triangles

Find the area of all the triangles as in the figure: (The first triangle from the right starts at 1, and going left they never end).

Solution

Compute the first few areas, with being the area of the largest triangle.

Evidently, this is a geometric series where .

Write out formula for sum of geometric series.

Compute sum.