So far, we have been learning about sequences. Today, we will take this further to learn about series. But before I do that, let me show the answers to the questions I posted yesterday.
1) 131072/5 2) 6 3) 12
Now to series. What exactly is a series? No, it is not a series of books, but rather it is the sum of all the terms in a sequences. The series of an infinite arithmetic sequence, such as 1, 2, 3…, is impossible to calculate, but it is always possible to find the series of a finite sequence. One simple but time-consuming way to solve for the series is to simply add up the terms one by one. But this doesn’t make math beautiful; in fact, it makes it seem clumsy. Rather, however, in certain types of sequences there are beautiful formulas that can be used to calculate a series.
Mathematician Carl Gauss was the first one to find a formula for an arithmetic series. Legend says that in his second grade class, he and his class was given the problem 1+2+3…+98+99+100=?. This is essentially just finding the series of the arithmetic sequence 1,2,3….98,99,100. While all his classmates were trying to solve this problem the stupid way, genius Gauss solved it in three seconds. How? Well, let’s think about it. If we took a1 which is 1 and summed up it with the last term, 100, we get 101. If we took the second term 2 and added it up to the second to last number 98 we also get 101. We can keep on adding the terms like this, until we approach to the very center. By the time we do that, we will have 50 101s. So, to get the total sum do 50 (101)=5050. Wala, this is the series!
So, from here, we can assume a basic formula for an arithmetic series, which is given a is a term and n is the number of terms and d is the constant difference
This formula happens to not only work for even number of terms but also for odd number of terms.
Well, that was the formula for an arithmetic series. How about a geometric series? Well, from our work with sequences, we know that a general geometric series would be:
a+a*r+a*r^2+…+a*r^(k-2)+a*r^(k-1) for k terms. Let this equal S or series. The next terms would be a*r^k, so if we add it to both sides:
a + a *r + a* r^2 +…+ a* r^(k−2) + a* r^(k−1) + a* r^k = S + a* r^k (Now move a to the other side)
a *r + a* r^2 +…+ a* r^(k−2) + a* r^(k−1) + a* r^k = S + a* r^k – a (Now factor an r at the left side)
r (a + a* r +…+ a* r^(k−3) + a* r^(k−2) + a* r^(k−1))= S + a* r^k − a [Now notice that the sum within the parentheses is equal to S, so..]
r*S= S + a* r^k − a [Now simply solve for S using algebra to get…]
Magic! This is the formula for the geometric finite series. How about geometric infinite series? Well, it turns out the only type of infinite series you can solve are infinite geometric whose terms’ absolute value keep on going lower and lower. One example would be the series of the sequence 1/2. 1/4. 1/8…. Here, the sequence approaches 0 but never meets it. However, to use the previous formula to solve this kind of series, you should substitue (1-r^n) with 0, thus the formula being a/(1-r).
Well, that’s it for series. If you want to know more, I suggest more research on y0ur own. For my next post, I will perhaps talk and discuss about a short story that I will introduce tomorrow.