Infinite Sequences help

[dunning2014]

I have several questions with answers, but not sure how they were arrived at. Any help in giving details of those answers will be greatly appreciated. See attached images for questions and answers. Thank you.

 

[Ishuda]

I have several questions with answers, but not sure how they were arrived at. Any help in giving details of those answers will be greatly appreciated. See attached images for questions and answers. Thank you.

In order:
a_n = n¹⁰ / 2ⁿ
Now 2ⁿ = (2^n/10)¹⁰, \(\displaystyle e^{ln(a)} = a\) and \(\displaystyle ln(a^x) = x ln(a)\) so
2^n/10 = \(\displaystyle e^{ln(2^{\frac{n}{10}})} = e^{\frac{n}{10} ln(2)} \)


Second: Multiply numerator and denominator \(\displaystyle \pi\)/n. Then the 2\(\displaystyle \pi\) can come outside the limit because it is constant.

Third: See above for part of it and for the other part log(a) - log(b) = log(a/b).

You should learn these rules as you will likely need them in the future.

 

[dunning2014]

Thanks!