Sequences And Series

We are given two sequences: and We need to compare these sequences and determine the relationship between them for all .

Step 1: Understanding
The sum of squares of the first natural numbers is given by: This expression grows roughly as for large , which means that increases cubicly as increases.

Step 2: Understanding
The sequence is given by: This grows much faster than , since grows exponentially and grows faster than any polynomial function. To get a rough idea of the growth rate of , recall that grows as (by Stirling's approximation), and multiplying this by gives an even faster growth rate.

Step 3: Comparing and
Clearly, grows much faster than . For large , the exponential growth of will always outpace the cubic growth of . To illustrate, let’s check the first few values of and : - For : So . - For : So . - For : So . - For : So . As we can see, for every , grows much faster than .

Step 4: Conclusion
From the calculations and growth rates, we can conclude that:

Answer: