Concrete mathematics : a foundation for computer science

64 SUMS
18 Let 9%~ and Jz be the real and imaginary parts of the complex number
z. The absolute value Iz/ is J(!??z)~ + (3~)~. A sum tkeK ok of complex
terms ok is said to converge absolutely when the real-valued sums
t&K *ak and tkEK ?ok both converge absolutely. Prove that tkEK ok
converges absolutely if and only if there is a bounding constant B such
that xkEF [oki < B for ,a11 finite subsets F E K.
Homework exercises
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Use a summation factor to solve the recurrence
To = 5;
2T,, = nT,-, + 3 . n! , for n > 0.
Try to evaluate ~~=, kHk by the perturbation method, but deduce the
VdUe of ~~=:=, Hk instead.
Evaluate the sums S, = xc=o(-l)n-k, T, = ~~=o(-l)n-kk, and Ll, =
t;=o(-l)n-kk2 by the perturbation method, assuming that n 3 0.
Prove Lagrange’s identity (without using induction): It’s hard to prove
the identity of
t (Cljbk-Clkbj)2 = (~Cl~)(~b~) - (LClkbk)‘.
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