• Title: Series - Introduction

  • Series: Real Analysis

  • Chapter: Infinite Series

  • YouTube-Title: Real Analysis 15 | Series - Introduction

  • Bright video: https://youtu.be/BgfP3riDcrc

  • Dark video: https://youtu.be/wwVHxHJOG_4

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ra15_sub_eng.srt

  • Timestamps

    00:00 Intro

    00:19 Introducing series

    01:07 Example of a series

    02:46 Definition series

    04:18 Rewriting the previous example

    05:04 Another example

    05:48 Credits

  • Subtitle in English

    1 00:00:00,529 –> 00:00:03,528 Hello and welcome back to real analysis

    2 00:00:04,257 –> 00:00:08,025 and first I want to thank all the nice supporters on Steady or Paypal.

    3 00:00:08,225 –> 00:00:13,506 In today’s part 15 we will talk about infinite sums, also called series.

    4 00:00:13,706 –> 00:00:19,675 Indeed this is such an important topic that we will use the next videos to talk about all the details.

    5 00:00:20,043 –> 00:00:25,581 Here we just start how we can define infinite sum in a rigorous way

    6 00:00:25,986 –> 00:00:30,762 and in the end you will see this is not so new. A series is just a special sequence

    7 00:00:31,143 –> 00:00:34,612 and how you deal with sequences, you have already learned here.

    8 00:00:35,443 –> 00:00:41,126 Ok, now in some problems it can occur that we need to add up infinitely many numbers.

    9 00:00:41,500 –> 00:00:48,463 For example we have the number a_1, then we add a_2, then a_3, a_4 and so on.

    10 00:00:49,014 –> 00:00:57,029 To make this shorter you could say, let’s use the sum symbol, the capital Sigma, where we go from k = 1 to infinity.

    11 00:00:57,229 –> 00:01:02,376 This is then what we call a series. So just adding infinitely many numbers.

    12 00:01:02,814 –> 00:01:07,342 So you see this is not so complicated. So let’s immediately look at an example.

    13 00:01:07,886 –> 00:01:11,876 The only thing we need here is a sequence a_k of real numbers.

    14 00:01:12,429 –> 00:01:17,548 Therefore let’s take one we already know. Which is -1 to the power k.

    15 00:01:18,114 –> 00:01:21,975 So this is a well defined sequence. However not a convergent one.

    16 00:01:22,271 –> 00:01:26,768 Nevertheless we still could ask: “What is the infinite sum of this sequence?”

    17 00:01:27,471 –> 00:01:31,271 Here we know, we start wit -1 and then we add 1

    18 00:01:31,471 –> 00:01:34,562 and then afterwards we add -1 again

    19 00:01:35,243 –> 00:01:38,500 and then 1 again, then -1 again and so on.

    20 00:01:38,700 –> 00:01:47,854 Of course this is what we can easily calculate, because we can set parentheses here and here and then we see, we just add zeros.

    21 00:01:48,471 –> 00:01:52,604 Hence the result of this infinite sum should be 0 as well.

    22 00:01:53,086 –> 00:01:57,741 Ok, at this point you should ask: “Why do we set the parentheses in this way?”

    23 00:01:57,941 –> 00:02:00,892 Of course there are also other possible ways.

    24 00:02:01,529 –> 00:02:06,917 For example we could skip the first element -1 and set the parentheses here.

    25 00:02:07,117 –> 00:02:11,262 Then we still add zeros, but what remains is -1.

    26 00:02:11,657 –> 00:02:15,345 Ok, now we have 2 different possible results.

    27 00:02:15,829 –> 00:02:19,987 So we immediately see such infinite sums here make problems.

    28 00:02:20,187 –> 00:02:24,288 So we need to define them and they don’t act like normal sums at all,

    29 00:02:25,129 –> 00:02:28,794 because in a normal sum we can set the parentheses as we want

    30 00:02:29,571 –> 00:02:34,895 and at the moment we don’t know in which cases we are allowed to this for an infinite sum as well.

    31 00:02:35,186 –> 00:02:41,409 This might be a little bit confusing, because we use the same symbol, but we could have different calculation rules.

    32 00:02:42,286 –> 00:02:46,479 Therefore first let’s define the symbol in a mathematical way.

    33 00:02:46,986 –> 00:02:49,627 So this will be our definition of a series.

    34 00:02:50,400 –> 00:02:54,022 For this let a_k be any sequence of real numbers

    35 00:02:54,486 –> 00:03:03,874 and then we define a new sequence s_n by setting s_n to be the sum of the first n members of the sequence a_k.

    36 00:03:04,343 –> 00:03:07,465 So this is just a normal finite sum.

    37 00:03:07,665 –> 00:03:10,263 Hence s_n is a real number as well.

    38 00:03:10,463 –> 00:03:15,063 Therefore the whole sequence given by s_n is what we call the series

    39 00:03:15,514 –> 00:03:21,558 and now you should see, in the case that this sequence is convergent we have a meaning for the infinite sum.

    40 00:03:21,957 –> 00:03:27,587 Therefore we are also able to define the symbol, where we have infinity at the sum here.

    41 00:03:28,129 –> 00:03:32,420 So this is simply the limit n to infinity of s_n

    42 00:03:32,986 –> 00:03:38,387 or without using a new name you would simply say it’s the limit of these partial sums.

    43 00:03:38,587 –> 00:03:42,361 Ok, there we have the case that often leads to confusion.

    44 00:03:42,561 –> 00:03:46,567 Namely this symbol here is frequently used in 2 different meanings.

    45 00:03:47,243 –> 00:03:51,235 On the one hand you can use it for the whole sequence s_n,

    46 00:03:51,435 –> 00:03:54,370 because then you don’t need to introduce a new name

    47 00:03:54,570 –> 00:03:59,228 and on the other hand it’s also used for the limit, if it exists.

    48 00:03:59,428 –> 00:04:04,953 Therefore please be careful. Sometimes you are allowed to calculate with this symbol as a number,

    49 00:04:05,471 –> 00:04:09,066 but other times such a calculation can lead to contradictions.

    50 00:04:09,543 –> 00:04:16,000 Ok, so that’s the definition and the introduction for a series and in the next video I will show you an example.

    51 00:04:16,117 –> 00:04:18,786 Namely the important geometric series.

    52 00:04:19,271 –> 00:04:22,941 Hence a good transition would be to rewrite our example from above.

    53 00:04:23,329 –> 00:04:26,372 Let’s formulate this into the sequence of partial sums.

    54 00:04:27,000 –> 00:04:30,286 There a_k is -1 to the power k.

    55 00:04:30,886 –> 00:04:34,618 Now let’s write the sum here as a sequence with index n.

    56 00:04:34,818 –> 00:04:41,227 There for calculating the first member we put in the index n = 1 and get out -1.

    57 00:04:41,427 –> 00:04:48,400 Then for the next member we have n = 2. Which means we have -1 + 1 and we get 0

    58 00:04:49,086 –> 00:04:52,546 and then for n = 3 we get -1 again.

    59 00:04:53,271 –> 00:04:56,102 Then this simply repeats with this pattern

    60 00:04:56,571 –> 00:04:59,650 and here you know, this sequence is not convergent.

    61 00:05:00,229 –> 00:05:04,421 Ok, maybe for closing this video also another example would be helpful.

    62 00:05:04,700 –> 00:05:10,345 Ok, here I want to ask the question: “What happens when we put in +1 instead of -1?”

    63 00:05:11,000 –> 00:05:15,830 Of course then the power k does not matter at all, because we always get out 1.

    64 00:05:16,200 –> 00:05:22,852 Therefore for our sequence of partial sums we get out 1, then 2, then 3 and so on.

    65 00:05:23,143 –> 00:05:27,154 Obviously this is also a sequence that is not convergent.

    66 00:05:27,543 –> 00:05:30,880 However we could say it’s divergent to infinity.

    67 00:05:31,257 –> 00:05:36,137 Now I already told you that in the next video we will talk about the geometric series.

    68 00:05:36,857 –> 00:05:42,257 This will generalize these 2 examples, but then we will also get convergent series.

    69 00:05:42,859 –> 00:05:46,371 Therefore I hope I see you there and have a nice day! Bye!

  • Quiz Content

    Q1: Is the following sequence convergent? $$ (-1,0,-1,0,-1,0,-1, \ldots) $$

    A1: Yes, it is.

    A2: No, it isn’t, but it has two accumulation values.

    A3: No, it isn’t, but it has no accumulation values.

    Q2: Is it possible to define the series $\displaystyle \sum_{k = 1}^\infty a_k$ for any sequence of real numbers $(a_k)_{k \in \mathbb{N}}$?

    A1: Yes, $\displaystyle \sum_{k = 1}^\infty a_k$ stands for the sequence of partial sums.

    A2: No, there are exceptions.

    Q3: If the series $\displaystyle \sum_{k = 1}^\infty a_k$ is convergent, we use the same symbol to denote another value as well. Which one is it?

    A1: In this case, $\displaystyle \sum_{k = 1}^\infty a_k$ also stands for the limit of $(a_k)_{k \in \mathbb{N}}$.

    A2: In this case, $\displaystyle \sum_{k = 1}^\infty a_k$ also stands for the limit of $\displaystyle \left( \sum_{k = 1}^n a_k \right)_{n \in \mathbb{N}}$.

    A3: In this case, $\displaystyle \sum_{k = 1}^\infty a_k$ also stands for the the symbol $\infty$.

    A4: In this case, $\displaystyle \sum_{k = 1}^\infty a_k$ also stands for the the symbol $-\infty$.

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