24.
∑︁n=1∞n2n(n!)2
n→∞limnnn!=n→∞limn⋅n...n1⋅2⋅3...(n−1)n=0
so,
n→∞limnun=n→∞limnn2n(n!)2=0<1
so, series converges
25.
∑︁n=1∞2n+1n3+1
using L'Hopital's rule:
n→∞lim2n+1n3+1=n→∞lim2nln326=0
then:
n→∞limnun=n→∞limn2n+1n3+1=0<1
so, series converges
26.
2⋅3⋅41+3⋅4⋅52+4⋅5⋅63+5⋅6⋅74+...=∑n=1∞(n+1)(n+2)(n+3)n
since (n+1)(n+2)(n+3)n<n21 and series ∑n21 converges, then series
∑n=1∞(n+1)(n+2)(n+3)n converges as well
27.
33−12−1+43−13−1+53−14−1+...=∑n=1∞(n+1)3−1n−1
un=(n+1)3−1n−1,vn=n5/21
n→∞lim(un/vn)=1=0
we know that series ∑n=1∞n5/21 converges, so series ∑n=1∞(n+1)3−1n−1 converges as well
28.
1p2+2p3+3p4+...=∑n=1∞npn+1
un=npn+1,vn=np−11
n→∞lim(un/vn)=1=0
then:
since series ∑n=1∞np−11 diverges if p ≤ 2 and converges if p > 2,
same thing for ∑n=1∞npn+1 : diverges if p ≤ 2 and converges if p > 2
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