What Went Wrong?

Calculus Level 2

Is the following sequence of steps valid? If no, which step is wrong?

Step 1: Consider the following limit:

L=limn1+2+3++nn2 L = \lim_{n \rightarrow \infty} \frac { 1 + 2 + 3 + \cdots + n } { n^2 }

Step 2: Verify that it is of the indeterminate form \dfrac { \infty}{\infty} .

  • Numerator tends to infinity
  • Denominator tends to infinity

Step 3: Apply L'hôpital's rule to get

L=limn12n=0  . L = \lim_{n \rightarrow \infty} \frac{ 1 } { 2n } = 0 \; .

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