5.Σ(k=1,3)(-1)^(k+1) sin (π/k)
=sin π-sin π/2+sin π/3
=0-1+√3/2
=(-2+√3)/2
15.1-1/2+1/3-1/4+1/5
=Σ(k=1,5) (-1)^(k+1) 1/k
[(-1)^(k+1) 只是用來控制正負號，k為奇數時為1偶數時為-1]
25.Σ(k=1,5) k(3k+5)
=Σ(k=1,5) (3k^2+5k)
=3Σ(k=1,5) k^2 + 5Σ(k=1,5) k
=165+75
=240
35.
(a).Σ(k=1,4) sin [-π+(k-1)(π/2)] (π/2)
=(π/2)Σ(k=1,4) sin [-π+(k-1)(π/2)]
=(π/2)[sin (-π)+sin (-π/2)+sin 0+sin (π/2)]
=(π/2)(0-1+0+1)
=0
(b).Σ(k=1,4) sin (-π+kπ/2) (π/2)
=(π/2)Σ(k=1,4) sin (-π+kπ/2)
=(π/2)[sin (-π/2)+sin 0+sin π/2+sin π]
=(π/2)(-1+0+1+0)
=0
(c).Σ(k=1,4) sin [(2k-5)π/4] (π/2)
= (π/2)Σ(k=1,4) sin [(2k-5)π/4]
=(π/2)[sin (-3π/4)+sin (-π/4)+sin π/4+sin 3π/4]
=(π/2)(-1/√2 -1/√2 +1/√2 +1/√2)
=0
45.△Xi=1/n，Ck=k/n
Σ(k=1,n) 2(k/n)^3 (1/n)
=Σ(k=1,n) (2k^3)/(n^4)
lim(n→∞) Σ(k=1,n) (2k^3)/(n^4)
=lim(n→∞) (2/n^4)[n(n+1)/2]^2
=lim(n→∞) (2/n^4)(n^2)[(n+1)^2]/4
=lim(n→∞) (1+1/n)^2 / 2
=1/2
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