Comment on Activity 1

We show the graph of `f text[(] x text[)] = 2^xtext[/]x` in Figure A1.

Figure A1   Graph of `ftext[(] x text[)] = 2^xtext[/]x`

This graph suggests that the function has a local minimum between `1` and `2`, and thereafter its values get steadily bigger as we go to the right. Furthermore, it appears that the graph is always concave upward, which means it can't level out as `x` gets large. Thus `f text[(] x text[)] rarr oo` as `x rarr oo`. Now the values of `2^m//m` are values of `f` at integer values of `x`, so we conclude that

`lim_(m rarr oo) 2^m/m = oo`.

If we substitute `m=k+1` in this formula, it tells us that the terms of the series in part (c) grow without bound (in absolute value) as `k rarr oo`. Thus, after the first few terms, we are alternately adding and subtracting larger and larger numbers. The effect of this is to generate partial sums that oscillate wildly — in particular, sums that cannot possibly settle down to a limiting value. We show the first 10 of those sums in Table 1.

Table A1   Partial sums of `sum_( )(-1)^k2^(k+1)text[/](k+1)`
k kth term kth sum
0
2
2
1
-2
0
2
2.667
2.667
3
-4
-1.333
4
6.4
5.067
5
-10.667
-5.6
6
18.286
12.686
7
-32
-19.314
8
56.889
37.575
9
-102.4
-64.825

What you should have observed by now in our examples of infinite series is that there is no hope of getting a finite sum from infinitely many terms unless those terms get small. That is, in order for `sum_( ) b_k` to converge, we must have `b_k rarr 0` as `k rarr oo`. Since the terms of the series in Activity 1 do not approach zero, the series cannot have a finite sum.