Finding lim n → ∞ 1 + 2 2 + 3 3 + 4 4...

aggierabz2006zw

aggierabz2006zw

Answered

2022-07-13

Finding lim n 1 + 2 2 + 3 3 + 4 4 + + n n n n
Finding
lim n 1 + 2 2 + 3 3 + 4 4 + + n n n n
Attempt:
lim n [ 1 n n + 2 2 n n + 3 3 n n + + n n n n ] = 1
because all terms are approaching to zero except last terms
but answer is not 1 , could some help me to solve it , thanks

Answer & Explanation

Giovanna Erickson

Giovanna Erickson

Expert

2022-07-14Added 14 answers

Bounding by a geometric series,
n n n n + ( n 1 ) n 1 n n + ( n 2 ) n 2 n n + + 1 1 n n 1 + 1 n + 1 n 2 + 1 n 3 + = n n 1
Since the sum is obviously always 1, and n n 1 , the Squeeze Theorem says that the limit is 1
Jamison Rios

Jamison Rios

Expert

2022-07-15Added 6 answers

By Stolz
lim n 1 + 2 2 + 3 3 + 4 4 + + n n n n = lim n n n n n ( n 1 ) n 1 = 1

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