Q. 52 Determine which of the limit of ... [FREE SOLUTION]

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Chapter 4: Q. 52 (page 326)

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value
$\lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} \frac{k^{3}}{n^{4} + n + 1}$

Short Answer

Expert verified

The limit of the sum is finite and it is equal to $\frac{1}{4}$ .

Step by step solution

Step 1. Given information

Given :

\lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} \frac{k^{3}}{n^{4} + n + 1}

Step 2. Find limit of the sum.

$\lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} \frac{k^{3}}{n^{4} + n + 1} = \lim_{n 鈫� 鈭�} \frac{1}{n^{4} + n + 1} {鈭�}_{k = 1}^{n} k^{3} = \lim_{n 鈫� 鈭�} \frac{1}{n^{4} + n + 1} {(\frac{n (n + 1)}{2})}^{2} = \lim_{n 鈫� 鈭�} \frac{1}{n^{4} + n + 1} (\frac{n^{2} {(n + 1)}^{2}}{4}) = \lim_{n 鈫� 鈭�} \frac{1}{n^{4} + n + 1} (\frac{n^{2} (n^{2} + 2 n + 1)}{4}) = \lim_{n 鈫� 鈭�} \frac{n^{4} (1 + \frac{2}{n} + \frac{1}{n^{2}})}{4 n^{4} (1 + \frac{1}{n^{3}} + \frac{1}{n^{4}})} = \lim_{n 鈫� 鈭�} \frac{(1 + \frac{2}{n} + \frac{1}{n^{2}})}{4 (1 + \frac{1}{n^{3}} + \frac{1}{n^{4}})} = \frac{1 + 0 + 0}{4 (1 + 0 + 0)} = \frac{1}{4}$

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91影视

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value
$\lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} \frac{k^{3}}{n^{4} + n + 1}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find limit of the sum.

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91影视

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value limn鈫�鈭�鈭�k=1nk3n4+n+1

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find limit of the sum.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Mechanics Maths

Pure Maths

Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value
$\lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} \frac{k^{3}}{n^{4} + n + 1}$