Nash Frank

2022-07-21

If I found that a series converges, how can I know to what number it's converging to?

I started learning series in calculus and I have trouble catching a basic concept. When I try to find if a series converges or diverges I have many ways to go about it. If I see that the series diverges than I stop there. If I see that the series converges than there is a number it's converging to right?

For example: $\sum \frac{2}{{n}^{3}+4}$. I do the limit comparison test with the series $\sum \frac{1}{{n}^{3}}$ and get a finite number 2. I know that $\sum \frac{1}{{n}^{3}}$ converges, so now I know that $\sum \frac{2}{{n}^{3}+4}$ converges also. How do I know to what number it converges to?

I started learning series in calculus and I have trouble catching a basic concept. When I try to find if a series converges or diverges I have many ways to go about it. If I see that the series diverges than I stop there. If I see that the series converges than there is a number it's converging to right?

For example: $\sum \frac{2}{{n}^{3}+4}$. I do the limit comparison test with the series $\sum \frac{1}{{n}^{3}}$ and get a finite number 2. I know that $\sum \frac{1}{{n}^{3}}$ converges, so now I know that $\sum \frac{2}{{n}^{3}+4}$ converges also. How do I know to what number it converges to?

decoratesuw

Beginner2022-07-22Added 11 answers

Step 1

Other than computing your series numerically (which is usually unsatisfying when it comes to representing a series in a "nice" way with convenient/aesthetic constants like $e,\pi ,i,\gamma ,$ etc.), the results of most convergence/divergence tests is usually just that: a truth value about whether the thing converges or diverges. For example, the Ratio Test almost always tells you about the nature of convergence, but nothing about it is designed to reveal what the series converges to.

Unfortunately, it is not trivial at all to determine what a series converges to. Before I introduce an example, it is worth our time to make sure everyone is on the same page. Out of all the types of series that you will encounter in your study of series, a "p-series" are probably the easiest ones to determine the nature of convergence/divergence. A p-series is in the form $\zeta (p)=\sum _{k=1}^{\mathrm{\infty}}\frac{1}{{k}^{p}}$.

Some of you should recognize that the above series is really just the Reimann zeta function. It is worth mentioning what happens to $\zeta (p)$ when p takes on certain values. If $p=1$, we have what is known as the Harmonic Series which has very wide-ranging applications, particularly in music/acoustics. It turns out that even though the value decrease for each term in $\zeta (1)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots $.

Step 2

Anyway, it can also be shown that the only values of p for which $\zeta (p)$ converges are when $p>1$. Now time for an example. Consider a problem (known as the Basel Problem) that stumped many people for many years (until Euler came along [as usual]): We can show readily that p-series converge when $p>1$. So a simple case is when $p=2$.

Other than computing your series numerically (which is usually unsatisfying when it comes to representing a series in a "nice" way with convenient/aesthetic constants like $e,\pi ,i,\gamma ,$ etc.), the results of most convergence/divergence tests is usually just that: a truth value about whether the thing converges or diverges. For example, the Ratio Test almost always tells you about the nature of convergence, but nothing about it is designed to reveal what the series converges to.

Unfortunately, it is not trivial at all to determine what a series converges to. Before I introduce an example, it is worth our time to make sure everyone is on the same page. Out of all the types of series that you will encounter in your study of series, a "p-series" are probably the easiest ones to determine the nature of convergence/divergence. A p-series is in the form $\zeta (p)=\sum _{k=1}^{\mathrm{\infty}}\frac{1}{{k}^{p}}$.

Some of you should recognize that the above series is really just the Reimann zeta function. It is worth mentioning what happens to $\zeta (p)$ when p takes on certain values. If $p=1$, we have what is known as the Harmonic Series which has very wide-ranging applications, particularly in music/acoustics. It turns out that even though the value decrease for each term in $\zeta (1)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots $.

Step 2

Anyway, it can also be shown that the only values of p for which $\zeta (p)$ converges are when $p>1$. Now time for an example. Consider a problem (known as the Basel Problem) that stumped many people for many years (until Euler came along [as usual]): We can show readily that p-series converge when $p>1$. So a simple case is when $p=2$.

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B)$6c{m}^{2}/s$

C)$12c{m}^{2}/s$

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EDIT

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