What changes once the terms are added instead of listed

0:00Take a sequence you already have, and write plus signs between its terms. One, three, five, seven becomes one plus three plus five plus seven. Nothing has been calculated. The plus signs are an instruction that has been written down and not yet carried out. That written-down instruction has a name of its own: it is a series. And it is a different object from the sequence it came from, even though not one number on the page has changed.

0:33The sequence answers the question what stands at position three. The series answers a different one: what is being added to what, and in which order. Carrying the addition out is a third thing again, and it is the one that produces a number. Almost every difficulty in this topic comes from letting the second and the third collapse into each other. So hold them apart deliberately. On the left, one plus three plus five plus seven: a series with four terms in it.

1:08On the right, sixteen: one number, with no terms at all. Sixteen is what the series comes to. It is not the series. Say it the other way round to hear how wrong it sounds: the series is sixteen terms long. Here are six different series, and every one of them comes to sixteen. The same four odd numbers; the same four turned round; sixteen on its own; four fours; two eights; and thirty-two with sixteen taken off it.

1:40Six series, seventeen terms between them, one number. Going from a series to its number loses almost everything and that is fine, because it is a different question. But it only goes one way, and no amount of staring at sixteen will tell you which series it came from. There is a sharper way to see the same thing, and it is worth doing carefully. Take four series and write each of them backwards.

2:10One plus three plus five plus seven becomes seven plus five plus three plus one. Three of the four came out as a genuinely different series, because a different term now stands first. The fourth did not, because it was five plus five plus five, and turning that round changes nothing. Now count how many of the four had their total changed by the turning. Nought. Not one of them.

2:40A series is an ordered thing and its total is not, and that single fact is enough to prove they cannot be the same object. If they were the same, changing one would have to change the other. Whether a series runs out is not something you decide when you write it. It is inherited from the sequence underneath, and there is no other source for it. Doubling from two, on ten positions, gives a series with ten terms: two plus four plus eight, all the way to one thousand and twenty-four.

3:17That one has a last term, so the addition can be carried out, and it comes to two thousand and forty-six. Now take the same rule on positions that never run out. Written down, the two look identical for as far as anyone is prepared to write. But asked for a total, the second one refuses, and it refuses for a reason you can state: there is no last term to stop at.

3:45Three endless series were written out five terms deep here and all three were refused a total. Not one of them was refused permission to be written. You can put a series on the page long before anybody knows whether it comes to a number. Written-out additions get long, so there is a shorthand, and the shorthand is where the confusion usually starts. A capital sigma, a letter underneath with where it starts, a number on top where it stops, and to the right, the thing being added.

4:21Read it as an instruction: take that expression, put one in for the letter, then two, then three, then four, and add up what you get. That is not an instruction to produce a number. It is a compact way of writing down the addition, which is exactly what a series is. Take the sigma with two times the letter, less one, running from one to four. Expand it and you get one plus three plus five plus seven.

4:51Not a number that happens to equal sixteen: that series, term for term, in that order. The sigma and the written-out version are two ways of writing the same object. One of them fits on a line and the other does not. The letter underneath is worth a minute, because students routinely think it is part of the answer. It is not. It is internal to the expression, and nothing outside can see it.

5:22The same sigma was rewritten here with five different letters underneath. The number of them that changed the series is nought. But there is a condition, and it is the thing to actually understand. You have to change the letter in both places: underneath the sigma, and inside the expression to the right. Change it underneath only, and what you have left is not the same series written differently, it is broken.

5:51All five of those refused, because the expression now mentions a letter nobody has said anything about. The number on top is a different matter entirely. Six different numbers on top gave series of one, two, three, four, eight and twelve terms, so that one is not internal at all. It is worth being able to go both ways, because reading is easier than writing and only one of them is examined.

6:23Unfolding first. The letter times itself, running from one to five, unfolds to one plus four plus nine plus sixteen plus twenty-five. Take each of the numbers one to five in turn, multiply it by itself, and write the results down with plus signs between them. Now fold one up. You are given the written-out addition and you have to find the rule that makes each term from its position.

6:54The first term must come out when the letter is one, and the last when it is five. That is what pins the number on top. And if you carry the addition out, that series comes to fifty-five. One more number in the world, and the series is still sitting there unchanged. Here is the ordinary shape of a question on this. You are given a rule that leans on the term before: start at one, and each term afterwards is two more than the one before it.

7:27First produce the terms. One, three, five, seven, nine. Then, and only then, write the series: one plus three plus five plus seven plus nine. Doing it in the other order is not possible, because the series is written from the terms. That series comes to twenty-five. And if you stop the addition part way, you get a shorter series, not a new kind of object. Stopping after one term, then two, then three, and so on gives one, four, nine, sixteen, twenty-five.

8:02Every one of those is a whole number times itself, and it keeps happening: out to twenty terms, not one running total fails it. And notice the fourth of them: stopping after four terms hands you back the series this video started with, and the number sixteen. The expression to the right of a sigma can be anything you can evaluate at a position. Take two plus three to the power of the letter, running from one to eleven.

8:34Eleven terms, the first three being five, eleven and twenty-nine. Nothing about that is harder than what came before. It is still a series, and it is written down, not worked out. It does happen to split usefully. The two, contributing once for each of the eleven positions, accounts for twenty-two. The powers of three account for two hundred and sixty-five thousand, seven hundred and nineteen. Together, two hundred and sixty-five thousand, seven hundred and forty-one.

9:10Splitting it left the total alone, and it did not leave the series alone: the three of them are three different series. How you get that middle number without adding eleven things by hand is a later question, and it has a good answer. Two last things a series does not promise you. It does not promise that the terms grow. Start at two, then two again, and after that each term is one less than the one before: two, two, one, nought, minus one.

9:44The running totals go two, four, five, five, and then back down to four. Adding is bookkeeping, not accumulating, and the arithmetic does not mind. And a series does not promise a number at the end of it. A half, then a quarter, then an eighth, then a sixteenth, then a thirty-second, and on for ever. That series is written down completely, in the sense that the rule is fixed and every term is determined.

10:14Whether it comes to anything is a genuine question, and it is not answered by anything said so far. It does have an answer, and getting to it needs one more idea, which is what the next few topics are for.