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,

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the answer you are looking for is a..

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a line is an defined derm because it

a)contains an infinite number of points b) can be used to create other geometric shapes c)describes something that does not have width or depth d) is a term that does not have a formal definition. a line is an undefined term because it: contains an infinite number of points.