2n+1 例5证明级数∑ 是收敛的, =1n(n+ 证因 2i+1 ∑ 2(+1) n lims.=lim 1 n→)00 (n+1) 即 2n+1 n2(n+1)例5 证明级数 是收敛的， ( ) 2 2 1 2 1 n 1 n n n ∞ = + + ∑ 证 因 ( ) ( ) ( ) 2 2 2 2 2 1 1 2 1 1 1 1 1 1 1 1 n n i n i s i i i i n ∞ = = ⎛ ⎞ + = = ⎜ ⎟ − = − ⎜ ⎟ + + + ⎝ ⎠ ∑ ∑ ( ) 2 1 li m l i m 1 1 1 n n n s n →∞ →∞ ⎛ ⎞ ∴ = − ⎜ ⎟ = ⎜ ⎟ + ⎝ ⎠ ( ) 2 2 1 2 1 1. n 1 n n n ∞ = + = + ∑ 即