Solving a Logarithmic Integral: Is it Right?

[Alem2000]

I don't know what's wrong with me...I have been studing for 6 hrs straight...and i noticed I have been making stupid mistakes...so can someone tell me what's up with my work...this is what I did. $$\sum_{n=2}^{\infty}\frac{1}{nln(n)}$$ must show if it converges or diverges soooo this is what i did

$$\int_{2}^{\infty}\frac{1}{nln(n)}dn$$ integration by parts and

$$u=\frac{1}{ln(n)}$$ ,
$$du=\frac{1}{n(ln(n))^2}$$ ,
$$v=ln(n)$$ ,
$$dv=\frac{1}{x}$$

$$\int_{2}^{\infty}\frac{1}{nln(n)}dn$$=$$\frac{ln(n)}{ln(n)}-\int_{2}^{+\infty}\frac{ln(n)}{nln(n)^2}dn$$

my logarithms cancel out and become $$1$$ and my integral of logorithms the top cancels and becomes one and i add it to the other side divide by two and i get $$\frac{1}{2}$$ now which say it converges did I do that right?

 

[Alem2000]

darn it i realized what I have been doing wrong!

bad integration i should have done $$u=ln(n)$$
$$du=\frac{1}{n}dx$$
plug and chug back into $$\int_{2}^{\infty}\frac{1}{nln(n)}dn$$ and I got$$\int_{2}^{\infty}\frac{1}{nu}ndu$$ which becomes $$\int_{2}^{\infty}\frac{1}{u}du$$ and eazy sailing after that ay?

 

[M98Ranger]

First of all, it's completely normal to make mistakes while studying for long periods of time. It's important to take breaks and give your brain a rest. Now, let's take a look at your work.

Your integration by parts is correct, but your final answer is not. When you cancel out the logarithms, you should have \frac{1}{n} left in the integral, not 1. Also, when you add \frac{1}{2} to the other side, you are essentially saying that the integral converges to \frac{1}{2}, which is not correct.

To determine if the integral converges or diverges, you need to evaluate the integral using a limit. As n approaches infinity, the integral will approach a certain value, and depending on that value, you can determine if the integral converges or diverges. In this case, the integral will approach 0, which means it converges.

So, in conclusion, your integration by parts is correct, but your final answer and reasoning are not. Remember to always use a limit when determining convergence or divergence of an integral. Keep practicing and don't be discouraged by mistakes!