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-√

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．
【正确答案】：(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)∵lim_n→∞=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