用比较审敛法判别下列级数的收敛性：
(1)∑n=1∞1/√n
(2)∑n=1∞1/(n2+2n+1)
(3)∑n=1∞(√n4+1-√n4-1)
(4)∑n=1∞∑sinπ/2n．
(1)∵1/√n﹥(1/n)而∑n=1∞1/n发散∴∑n=1∞1/√n发散． (2)∵1/(n+1)2≤1/n2 ∑n=1∞1/n2收敛，∴∑n=1∞1/(n+1)2收敛． (3)∑n=1∞(√n4+1-√n4-1) =∑n=1∞2/(√n4+1+√n4-1) ∵2/(√n4+1+√n4-1)≤1/(√n4-1,而∑1/√n4-1志，收敛． ∴∑n=1∞(√n4+1-√n4-1)收敛． (4)∵limn→∞=sin(π/2n)/(π/2n)=1 ∴∑n=1∞π/2n)与∑n=1∞sin(π/2n)同敛散 而∑n=1∞π/2n收敛 ∴∑n=1∞sin(π/2n)收敛
