Short Answer

Determine which of the limit of sums in Exercises 47–52 are infinite and which are finite. For each limit of sums that is finite, compute its value
$\lim_{n \to \infty} \sum_{k = 1}^{n} (k^{2} + k + 1)$

The limit of the sum is infinite.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information

$\lim_{n \to \infty} \sum_{k = 1}^{n} (k^{2} + k + 1)$

Step 2. Find the limit of the sum.

$\lim_{n \to \infty} \sum_{k = 1}^{n} k^{2} + k + 1 = \lim_{n \to \infty} ({\sum k^{2}}_{k = 1}^{n} + \sum_{k = 1}^{n} k + \sum_{k = 1}^{n} 1) = \lim_{n \to \infty} (\frac{n (n + 1) (2 n + 1)}{6} + \frac{n (n + 1)}{2} + n) = \lim_{n \to \infty} (\frac{n (n + 1) (2 n + 1) + 3 n (n + 1) + 6 n}{6}) = \lim_{n \to \infty} (\frac{n (2 n^{2} + 3 n + 1) + 3 n^{2} + 3 n + 6 n}{6}) = \lim_{n \to \infty} (\frac{2 n^{3} + 3 n^{2} + n + 3 n^{2} + 3 n + 6 n}{6}) = \lim_{n \to \infty} (\frac{2 n^{3} + 6 n^{2} + 10 n}{6}) = \infty$

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Calculus

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Short Answer

Determine which of the limit of sums in Exercises 47–52 are infinite and which are finite. For each limit of sums that is finite, compute its value
$\lim_{n \to \infty} \sum_{k = 1}^{n} (k^{2} + k + 1)$

Step by Step Solution

Step 1. Given information

Step 2. Find the limit of the sum.

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Determine which of the limit of sums in Exercises 47–52 are infinite and which are finite. For each limit of sums that is finite, compute its value limn→∞∑k=1n(k2+k+1)

Step by Step Solution

Step 1. Given information

Step 2. Find the limit of the sum.

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Determine which of the limit of sums in Exercises 47–52 are infinite and which are finite. For each limit of sums that is finite, compute its value
$\lim_{n \to \infty} \sum_{k = 1}^{n} (k^{2} + k + 1)$