Question

Calculus 2 comparison theorem

Answer 1

@ganeshie8

Answer 2

Notice that

\(x^{1.1} + 5\ge x+ 5\) in \([1, \infty]\)

Answer 3

that means 1/(x^1.1+5) <= 1/x+5

Answer 4

so u can test and see if below integral converges :
\(\large \int \limits_1^{\infty} \frac{1}{x+5} dx\)

Answer 5

and clearly that does NOT converge.

Answer 6

where did u get [1, infinity]?

Answer 7

@ganeshie8

Answer 8

from the problem itself
look at the bounds of ur integral

Answer 9

\[
\frac{1}{x^{1.1}+5}<\frac{1}{x^{1.1}}
\\
\int \frac{1}{x^{1.1}} \, dx=-\frac{10.}{x^{0.1}}\\
\int_1^{\infty } \frac{1}{x^{1.1}} \, dx=10

\]
Your integral is convergent by the comparison theorem
@ganeshie8

Answer 10

In general
\[
\int_1^\infty \frac {dx}{x^p}
\]
is convergent if p>1 and
divergent if \( p\le 1\)

Answer 11

Thank you both of you