Combine an alternating GP with logarithms

(i) Find ∑r=1k[(−12)r+1+ln⁡(r+1)]\sum_{r=1}^k[(-\frac12)^{r+1}+\ln(r+1)] in terms of kk. Simplify your answer. marks

(ii) Hence determine if ∑r=1∞[(−12)r+1+ln⁡(r+1)]\sum_{r=1}^{\infty}[(-\frac12)^{r+1}+\ln(r+1)] exists. mark

[2022/HCI/Prelim/II/4a]

Total marks: 5

Solution

(i)

Split and sum

Find ∑r=1k[(−12)r+1+ln⁡(r+1)]\sum_{r=1}^{k}\left[\left(-\tfrac12\right)^{r+1}+\ln(r+1)\right] in terms of kk, simplified.

Explore this idea

Hint

  1. Separate the geometric and logarithmic parts.

  2. Sum the GP with its formula and combine the logs into one.

Answer · (i)

[1−(−1/2)k]/6+ln⁡((k+1)!)[1-(-1/2)^k]/6+\ln((k+1)!).

Explore the idea

A GP and a logarithm in one sum

Each bar is (−12)r+1+ln⁡(r+1)\left(-\frac12\right)^{r+1}+\ln(r+1). The logarithm dominates quickly; the GP is only a small wobble.

Upper limitr = 1 to 5
Try an example
Σ (−½)^(r+1) + ln(r+1), r = 1 to 5
6.751
Number of terms
5
⅙[1 − (−½)^k] + ln (k+1)!
6.751
matches the direct total
Key idea

Sums of logs become the log of a product, and sums of a GP use the GP formula. Handle each part separately.

Ideas to try
  1. Compare the direct total with the closed form at several kk. They always agree.
  2. Notice the first bar: 14+ln⁡2\frac14+\ln2. The GP part alternates in sign but shrinks.
(ii)

Limit of the partial sums

Hence determine whether ∑r=1∞[(−12)r+1+ln⁡(r+1)]\sum_{r=1}^{\infty}\left[\left(-\tfrac12\right)^{r+1}+\ln(r+1)\right] exists.

Explore this idea

Hint

  1. Let k→∞k\to\infty in each part.

Answer · (ii)

Diverges to +∞+\infty; no finite sum.

Explore the idea

One part settles, the other never does

The geometric part levels off at 16\frac16, while ln⁡[(k+1)!]\ln[(k+1)!] keeps climbing, and so does the total.

Number of termsk = 10
Try an example
GP part at k = 10
0.1665
ln (k+1)! at k = 10
17.5023
total Sₖ at k = 10
17.6688
Key idea

A convergent series needs every part of its partial sum to settle; one unbounded part is enough to make the whole diverge.

Ideas to try
Slide kk to the right. Which line flattens and which does not?
Takeaway

Separate, then judge each part

To decide whether an infinite sum exists, find the partial sum in closed form and take the limit of each piece.