Lim n→∞[(n + n² + n³ + . nⁿ)/(1ⁿ + 2ⁿ + 3ⁿ +.nⁿ)]

Step-by-step explanation:

Given expression is

To, evaluate this limit, let we simplify numerator and denominator individually.

So, Consider Numerator

Clearly, if forms a Geometric progression with first term n and common ratio n respectively.

So, using Sum of n terms of GP, we get

Now, Consider Denominator, we have

can be rewritten as

Now, Consider

So, on substituting the values evaluated above, we get

Now, we know that,

So, using this, we get

Now, in denominator, its an infinite GP series with common ratio 1/e ( < 1 ) and first term 1, so using sum to infinite GP series, we have

Hence,

i think the answer is b)

step-by-step explanation: