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Chapter 7: Q. 6 (page 655)

Fill in the blanks to complete each of the following theorem statements.
Basic Limit Rules for Convergent Sequences: If $\{a_{k}\} and \{b_{k}\} are convergent sequences with a_{k} 鈫� L and b_{k} 鈫� M as k 鈫� 鈭�$ and if c is any constant, then
$a_{k} b_{k} 鈫�$

Short Answer

Expert verified

The required answer is $a_{k} b_{k} 鈫� L M$

Step by step solution

01

Step 1. Given Information

The given data is if $\{a_{k}\} and \{b_{k}\} are convergent sequences with a_{k} 鈫� L and b_{k} 鈫� M as k 鈫� 鈭�$

02

Step 2. Explanation

As sequences are also coined as functions, So, the properties of limits that applied for functions can also be applied to sequences.

Thus, $a_{k} b_{k} 鈫� L M$

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Find the values of x for which the series ${鈭�}_{k = 0}^{鈭�} {(\frac{3}{x})}^{k}$ converges.

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