convergence or divergence

[obtuse flax]

Test the series whose general term is :

(n²+1)^½ - n

[graceful tundra]

1. Wait patiently for a helper to come along.
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[obtuse flax]

So according to me it's convergent but the solution refers this as a divergent series

Take n² common and out of the parenthesis and then expand the binomial expression. The result can be simplified to form the eq:

1/2n -1/8n³+1/16n⁵....
This should be a converging series right ?

[stark canopy]

Just to verify, is this what the general term is?

Because I am a bit confused

[craggy current]

@obtuse flax use the integral test to verify that it diverges

Verify that it is a monotonically decreasing function. Idk if its easy to find the integral but it works.

Maybe some other convergence test could be used as well.

[frosty dirge]

im pretty sure u can use the integral test here

[frosty dirge]

oh yup

[obtuse flax]

Yess

[obtuse flax]

Oh I haven't learnt it yet

[frosty dirge]

if u know what integrals r its rlly simple

[craggy current]

@obtuse flax ok what about the limit comparison test where an=√(n²+1)-n and bn=1/n. You know that both an, bn are positive and you have the limit n→∞ an/bn as n→∞ n(√(n²+1)-n)

Multipling and dividing by √(n²+1)+n the limit becomes n/(√(n²+1)+n) pull out n from denominator and numerator you have 1/(√(1+1/n²)+1) which for n→∞ has limit 1/2. Therefore the limit exists and it is finite non zero. Since the series of bn=1/n (harmonic series) diverges then the series of an=√(n²+1)-n diverges as well

[frosty dirge]

if u dont use latex i am mentally unable to understand what ur saying but yh ig if u find the right sequence i would work

[obtuse flax]

Ah yes I know the comparison test using an auxiliary series

I'm sorry I found this confusing too but I understand a lil what you're explaining

[craggy current]

Well if you write it down on paper, i guess it will be more understandable

[obtuse flax]

Ohh okayy I understand now!

Thanks a lot @craggy current and @frosty dirge !!!

[rough forum]

@obtuse flax has given 1 rep to @frosty dirge @craggy current

[frosty dirge]

i mean i didnt necessarily help much but ok lol

[obtuse flax]

Still I appreciate your time and attention