Calculus 3 (Series) UNSOLVED PROBLEM

In summary, the conversation involves a user asking for help with a problem set and an editor requesting the user to show their progress. The specific problem mentioned is to show that $\displaystyle H_n - \ln(n) \geq 0$, which requires showing that $\displaystyle H_n \geq \ln(n)$. The editor suggests evaluating $\displaystyle \int_1^{n+1} \frac{1}{x} \mathrm{d}x$ and drawing a relevant picture.
  • #1
matdac
1
0
i have attached the problem set.

I have done the first three problems but number 4 is very difficult.

Can someone help me out?

Thanks

View attachment 7411

[Editor's note: The PDF below contains the complete problem set from which #4 is as shown above.]
 

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  • #2
Hello matdac and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?
 
  • #3
matdac said:
i have attached the problem set.

I have done the first three problems but number 4 is very difficult.

Can someone help me out?

Thanks
[Editor's note: The PDF below contains the complete problem set from which #4 is as shown above.]

If you want to show that $\displaystyle \begin{align*} H_n - \ln{ \left( n \right) } \geq 0 \end{align*}$ then you need to show that $\displaystyle \begin{align*} H_n \geq \ln{ \left( n \right) } \end{align*}$.

Can you at least evaluate $\displaystyle \begin{align*} \int_1^{n+1}{ \frac{1}{x}\,\mathrm{d}x} \end{align*}$ and see WHY it might be important to draw the picture you have been told to?
 

1. What is the unsolved problem in Calculus 3 (Series)?

The unsolved problem in Calculus 3 (Series) is known as the convergence or divergence of infinite series. This problem involves determining whether a given infinite series will approach a finite limit (converge) or will increase without bound (diverge).

2. Why is the convergence or divergence of infinite series important in Calculus 3?

The convergence or divergence of infinite series is important in Calculus 3 because it allows us to determine the behavior of a series and whether it can be used to approximate a value or not. It also has real-world applications in areas such as physics, economics, and engineering.

3. What techniques are commonly used to solve problems related to the convergence or divergence of infinite series?

Some common techniques used to solve problems related to the convergence or divergence of infinite series include the comparison test, the ratio test, the root test, and the integral test. These tests help us determine whether a series is convergent or divergent.

4. Are there any unsolved problems related to the convergence or divergence of infinite series in Calculus 3?

Yes, there are still some unsolved problems related to the convergence or divergence of infinite series in Calculus 3. One example is the Riemann Hypothesis, which states that all non-trivial zeros of the Riemann zeta function lie on the critical line.

5. How can the convergence or divergence of infinite series be applied in real-world situations?

The convergence or divergence of infinite series can be applied in real-world situations in various fields such as physics, engineering, and economics. For example, in physics, it can be used to model physical phenomena and make predictions. In economics, it can be used to analyze financial data and make informed decisions.

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