1
Question
limn→∞⎡⎢
⎢⎣1n+1√n2−12+1√n2−22+....+1√n2−(n−1)2⎤⎥
⎥⎦.
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Solution
limn→∞⎡⎢
⎢⎣1n+1√n2−12+1√n2−22+.....1√n2−(n−1)2⎤⎥
⎥⎦=limn→∞⎡⎢
⎢
⎢
⎢
⎢
⎢⎣1n+1n√1−1n2+1n√1−(2n)2+.....1n√1−(1−1n)2⎤⎥
⎥
⎥
⎥
⎥
⎥⎦=limn→∞⎡⎢
⎢
⎢
⎢
⎢
⎢⎣1n⎡⎢
⎢
⎢
⎢
⎢
⎢⎣1+1√1−1n2+1√1−(2n)2+...1√1−(1−1n)2⎤⎥
⎥
⎥
⎥
⎥
⎥⎦⎤⎥
⎥
⎥
⎥
⎥
⎥⎦=limn→∞[1n(1+1+1+1......1(ntimes))]=limn→∞[1n×n]=1
limn→∞⎡⎢
⎢⎣1n+1√n2−12+1√n2−22+.....1√n2−(n−1)2⎤⎥
⎥⎦
=limn→∞⎡⎢
⎢
⎢
⎢
⎢
⎢⎣1n+1n√1−1n2+1n√1−(2n)2+.....1n√1−(1−1n)2⎤⎥
⎥
⎥
⎥
⎥
⎥⎦
=limn→∞⎡⎢
⎢
⎢
⎢
⎢
⎢⎣1n⎡⎢
⎢
⎢
⎢
⎢
⎢⎣1+1√1−1n2+1√1−(2n)2+...1√1−(1−1n)2⎤⎥
⎥
⎥
⎥
⎥
⎥⎦⎤⎥
⎥
⎥
⎥
⎥
⎥⎦
=limn→∞[1n(1+1+1+1......1(ntimes))]
=limn→∞[1n×n]=1
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