We have known the concept of infinity ever since we were very young. For most people, infinity would mean infinitely getting closer to something. Also, everyone would claim that they are familiar with addition. Thus, since we are familiar with addition as well as infinity, we can consider some very interesting problems.
To start off, we can consider the summation of 1+1/2+1/3+…until the denominator is infinitely large, which means the term is infinitely small. Yet, what happens if we kept adding the terms of the series? We would then have an infinite amount of terms, but the term would always approach 0. However, what will the sum be? Will it be infinity? Or will it approach a certain value? Who will win here, the infinity or infinitely small?
Take some time to think about that before you continue.
Alright, this is a simple question. The proof is below:
As we keep on writing the terms for this series, we can keep constructing an infinite amount of such inequalities. Therefore, the sum of this series is greater than the sum of infinite one half, which means the sum tends to infinity and the series diverges. We call this series the harmonic series.
This may indeed be a really easy problem for you. Yet, we can examine more about the harmonic series. Now, we want to make things more interesting. What if we eliminate all the terms, which has 7 in their decimal representation of the denominator. For instance, we won’t include the fraction 1/17 in the series now, as the decimal representation of denominator has number 7.
Well, this doesn’t seem like a big deal. We still have infinite terms. What about the sum of this deficient harmonic series now?
I’ll give you a few more seconds.
Do you have the answer now? Okay, still very simple: the deficient series would converge now, which means its sum would approach a certain number. Actually, intuitively speaking, as the number in denominator grows larger, it will have a higher and higher probability of being eliminated in this rule, which may tend to 1. Then, if we have a finite number of terms, each of which is also finite, we won’t get an infinite sum.
We can use a similar method to prove our conjecture:
Alright, now we can look at something that’s a lot more trickier now. How about we only include all the terms whose denominator is a prime. We all know that there’s an infinite number of primes. There’s also infinite terms in this series, whose terms tend to 0. Yet, from the previous two examples, we have shown that such sequence could converge or diverge. Then, what would happen to this series. Will it converge or diverge?
This may be a little trickier and require a bit more time.
Do you have the answer now? The answer is that this series converge! Here is a proof:
How about if the denominator is all the squares of the nature numbers, will it converge or diverge? We will leave this to the readers to prove it. Be careful with our hints: consider polynomials and trigonometry.