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Chapter 1: Q. 79 (page 89)

For any positive integerk, the following equation holds:

1+2+3+路路路+k=k(k+1)2. Use this fact to prove that for all k > 100, the value of the sum of the first k integers is greater than 5000. What does this have to do with the limit of a sequence of sums as k鈫�鈭�?

Short Answer

Expert verified

The given statement is proved. The limit of a sequence of sums as k鈫�鈭�will also be鈭�.

Step by step solution

01

Step 1. Given Information. 

The given equation is1+2+3+路路路+k=k(k+1)2.

02

Step 2. Prove. 

Let the function is f(k)=k(k+1)2.

Take the limit of the above function as k鈫�101.

limk鈫�101f(k)=limk鈫�101k(k+1)2limk鈫�101f(k)=101(101+1)2limk鈫�101f(k)=1011022limk鈫�101f(k)=10151limk鈫�101f(k)=5151

Similarly for the other larger values of k,the value of the function will be even larger.

Therefore, the limit of a sequence of sums as k鈫�鈭�will also be鈭�.

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Most popular questions from this chapter

For each limit statement in Exercises 41-44, use algebra to find 未or Nin terms of 蔚or M, according to the appropriate formal limit definition.

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