5.3 SUMS AND PRODUCTS 163When we are concerned about convergence, the first few terms do not matter at all. In
fact, the first few trillion terms do not matter! Don't forget this neat idea of shifting
the index of summation to suit our purposes.Problems and Exercises
5.3.6 Find a formula for the product of the terms of a
geometric sequence.
5.3.7 Find a formula for the sumand generalize.
5.3.8 Find a formula for--I I I
+--+ ... +.
1·2 ·3 2·3·4 n(n+ l)(n+2)
Can you generalize this?
5.3.9 Find a formula for
1 · 2·3+2·3·4+···+n(n+ 1)(n+2).Can you generalize this?
5.3. 10 Observe that l v'44 J = 6, l vi 4444 J = 66.
Generalize and prove.
5.3.11 (AIME 1983) For {I, 2, 3, ... ,n} and each of
its nonempty subsets a unique alternating sum is de
fined as follows: Arrange the numbers in the subset in
decreasing order and then, beginning with the largest,
alternately subtract and add successive numbers. (For
example, the alternating sum for {I, 2, 4, 6, 9} is 9 -
6+4 -2 + I = 6 and for {5} it is simply 5.) For each
n, find a formula for the sum of all of the alternating
sums of all the subsets.
5.3.12 Prove thatThis is just a fancy way of saying that if you consider
each x in U and write down a I whenever x lies in A,
then the sum of these I s will of course be the number
of elements in A.
5.3. 13 Find the sum 1. l! + 2· 2! +. .. + n. nL
5.3.14 Find a formula for the sum
n k
�I (k+ I)!·n
5.3. 15 Evaluate the product J] cos ( 2 k (J ) •n I
5.3.16 Find the sum � -I -.
g 20 gku
5.3. 17 (AIME 1996) For each permutation
al ,a2,a3, ... ,aw of the integers 1,2, ... , 10, form the
sumFind the average value of all of these sums.
5.3. 18 Try Problem 1.3.8 on page 9, if you haven't
done it already.
5.3.19 (Canada 1989) Given the numbers
1,2,2^2 , •.• ,2n-^1 , for a specific permutation
a = XI ,X 2 , ... ,Xn of these numbers we define
SI(a) = xl,S 2 (a) = XI + X 2 , ... and Q(a) =
SI(a)S 2 (a)···Sn(a). Evaluate };1/Q(a), where the
sum is taken over all possible permutations.
5.3.20 A 2-inch elastic band is fastened to the wall
at one end, and there's a bug at the other end. Ev
ery minute (beginning at time 0), the band is instan
taneously and uniformly stretched by I inch, and then
the bug walks I inch toward the fastened end. Will the
bug ever reach the wall?
5.3.2 1 Let S be the set of positive integers which do
not have a zero in their base-1O representation; i.e.,
S = {l,2, ... , 9, Il, 12, ... ,19,21, ... }.
Does the sum of the reciprocals of the elements of S
converge or diverge?
5.3.22 Example 5.3.5 on page 162 showed that
,(2) < 00. Use "massage" to show that, in fact, ,(2) <- Then improve your estimate further to show that
,(2) < 7/4. (The exact value for ,(2) is 1C^2 /6. See
Example 9.4.8 on page 349 for a sketch of a proof. )
5.3.23 (Putnam 1977) Evaluate the infinite product