8.5 Series of Products 503m = 1, and using the sequence {xk} instead of {ak}, show that the
n n
conclusion of Theorem 8.5.2 becomes: I: bkb..ak-l = anbn+l - L akb..bk.
k=l k=l- Prove Lemma 8.5.4 (b).
- Prove Lemma 8.5.5 (b).
- Prove Lemma 8.5.6.
- Prove that the convergence in Examples 8.5.7 (a) and (b) is not absolute,
except in (a) when tis an integral multiple of 1r.
(^00) sin( ks) cos( kt)
- Prove that k"fl k converges for all real numbers t.
- Work out the details of the proof of Theorem 8.5.8.
. ~ ksin^2 k - Investigate the convergence of L..,,(-1) -k-, as follows:
k=l
(a) Use a trigonometric identity for sin^2 k to express the nth partial sum
o f ht 1s. senes. as a com b' mat10n. o f L.,, ~ ~ (-l)k an d ~( L.,, -l)kcos2k ~.
k=l k=l
(b) Use a trigonometric identity to show that (-l)k cos2k = cos(n+2)k.(c) Apply Lemma 8.5.4 to show that 1~ cos(n + 2)kl ::; -1
-.
L.,, cos 1
k=l
(d) Apply Dirichlet's test to the second of the two series in (a).
(e) Apply all of the above to prove that the given series converges.
(f) Show that the convergence is not absolute.
Prove that absolute summability is weaker than square summability by
showing that
(a) an absolutely summable sequence must be square summable.
(b) :3 a square summable sequence that is not absolutely summable.
Prove that square summability is neither weaker nor stronger than summa-
bility (that is, neither condition implies the other) by finding
(a) a summable sequence that is not square summable.
(b) a square summable sequence that is not summable.
00
Prove that if { ak} is square summable, then L ~ converges absolutely.
k=l