- #1
Dustinsfl
- 2,281
- 5
$\sum\limits_{n = 1}^{\infty}\left(\sqrt{1 + n^2} - n\right)$
$$
\sqrt{1 + n^2} - n = \frac{1}{\sqrt{1 + n^2} + n}
$$
Now what?
$$
\sqrt{1 + n^2} - n = \frac{1}{\sqrt{1 + n^2} + n}
$$
Now what?
dwsmith said:$\sum\limits_{n = 1}^{\infty}\left(\sqrt{1 + n^2} - n\right)$
$$
\sqrt{1 + n^2} - n = \frac{1}{\sqrt{1 + n^2} + n}
$$
Now what?
Chris L T521 said:\[\frac{1}{\sqrt{1+n^2}+n}\sim \frac{1}{n}\]
So it makes sense to compare this to $\dfrac{1}{n}$. I'd suggest using the limit comparison test to do this and show that
\[\lim_{n\to\infty}\dfrac{\dfrac{1}{n}}{\dfrac{1}{ \sqrt{1+n^2} +n}}\rightarrow L\]
If the limit converges (i.e. $L<\infty$), both terms have the same behavior (in this case, the limit should converge, implying that both series diverge).
I hope this helps!
dwsmith said:Wouldn't it be easier than to say that $\frac{1}{n} > \frac{1}{\sqrt{1+n^2}+n}$ and the Riemann Zeta function only converges for the $\text{Re}(\sigma) >1$. Since $\text{Re} (\sigma) =1$, it diverges.
The purpose of exploring this sum is to better understand the behavior of the series and potentially find patterns or relationships that can be applied in other mathematical contexts.
This sum is related to various areas of mathematics, including calculus, number theory, and complex analysis. It can also be used to approximate the value of certain integrals and solve certain equations.
This sum has practical applications in fields such as physics, engineering, and economics. It can be used to model various phenomena, such as the behavior of electrical circuits and the motion of objects in space.
No, there are currently no known closed forms or exact solutions for this sum. However, there are various techniques and methods for approximating its value and studying its properties.
The sum can be used in problem-solving and mathematical proofs to demonstrate relationships between different mathematical concepts and to derive new results. It can also be used to simplify complex equations and expressions.