#Real Analysis Convergence of Series Q

155 messages · Page 1 of 1 (latest)

eager palmBOT
hoary vine
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Tbh I don't understand what you are doing. The last line is definitely a wrong statement though. Let a_n = (-1)^n/n

worldly linden
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is that no convergent as well

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@hoary vine

hoary vine
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And the harmonic series diverges obviously

worldly linden
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right yes i forgot

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thanks

hoary vine
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Tbf, the statement isn't wrong because you probably made an notation error. -1^n = -1 for all n

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But if you mean (-1)^n then it is

worldly linden
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yeah im wrong

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didnt mean to tag u

hoary vine
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Show that they are converging to the same value

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Then conclude that the whole series is

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*Disclaimer: Not sure if they want you to go this way, but it definitely works

worldly linden
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I think I see what u are getting at

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my problem was I didnt realize I could just make my partial sums include the (-1)^n

hoary vine
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You have to

worldly linden
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which would then be a1-a2+a3-.....+/an

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I got it now thank you

hoary vine
worldly linden
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sure

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one sec

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I took a different approach

hoary vine
worldly linden
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okay which part

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or all

hoary vine
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I think you can't really fix it, but I might be wrong

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The part where you went from |s_n+1-s_n| < epsilon/2 to |s_m-s_N| < epsilon/2 isn't shown

worldly linden
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what if m,n>N+1

hoary vine
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What does this show? |s_m-s_N| can and will probably be bigger then |s_N+1-s_N|? (at least for most sequences)

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Also why did you let n be even if you don't use it?

worldly linden
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idk i think i was thinking along your hint earlier

hoary vine
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This may not be the intended way, but it works 100%: Show that s_2n is monotonous and bounded. Same for s_n+1

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You know that monotonous and bounded is enough for convergence

worldly linden
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right

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okay ill give it a shot

hoary vine
worldly linden
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okay thanks

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will u be able to check in a little bit?

hoary vine
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I'll go to sleep

worldly linden
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o ok

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no problem

hoary vine
worldly linden
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o

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ill check agains that then

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if u send link

hoary vine
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I didn't search for it

worldly linden
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o

hoary vine
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I was still remembering it though. The result you're trying to show is called Leibnitz criteria

worldly linden
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Ah interesting

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ill look for it then

hoary vine
worldly linden
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i think im gonna try again tmw

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ill probs do a different section of the text today

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but thanks

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if ur fine with it ill message u tmw

worldly linden
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@hoary vine

hoary vine
worldly linden
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i touched up my early proof

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and fixe the mistakes

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would u take a look

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its fine if not

hoary vine
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Yeah

worldly linden
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one sec

hoary vine
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,rotate

dreamy sigilBOT
hoary vine
worldly linden
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yes on accident

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ill send

hoary vine
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Thx

worldly linden
hoary vine
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S_N+1 = (-1)^(N+1)*a_(N+1)+ s_N and if N then S_N+1 < S_N

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I suggest we do the proof together?

worldly linden
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okay

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so where did i go wrong

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i cant make sense of what u typed

hoary vine
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If N is even then then S_(N+1) < S_N

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So the inequality in the 5th line is not true

worldly linden
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isnt it tho

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becauze u will be subtracting

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from the preceding odd one

hoary vine
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Yes, that makes S_(N+1) smaller

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than S_N

worldly linden
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yes i see

hoary vine
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But here you are claiming S_(N+1) > S_N

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Ok, now define s_2n = -a1+a2-a3+...-a_(2n-1)+a_(2n)

hoary vine
worldly linden
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im stil hung up on what u were saying

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i think n+1 is bigger

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sn will end with a subtraction

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and n+1 will add something back

hoary vine
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If N is even then the last term will be an addition. Because you subtract if n is odd and add if it is even

worldly linden
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okay

hoary vine
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I'm thinking right now if you can actually save what you did 🤔

worldly linden
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if i just flip them

hoary vine
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Yes let's say S_(N+1) < S_N

worldly linden
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right

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and now its good?

hoary vine
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I think you'd need to further elaborate on why s_n is in between those two for n >= N

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Then it'd be good I suppose

worldly linden
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i was trying to where it says that the original a_n is decreasing

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so obv the absolute value of s_n where n>N is going to be less than S_N

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is that sensible?

hoary vine
worldly linden
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lmao

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okay

hoary vine
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So you need to elaborate a bit more and then everything works 🙂

worldly linden
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awesome thanks for the help

hoary vine
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That's a huge improvement though

worldly linden
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can I add you on discordd

hoary vine
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No

worldly linden
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lol k

hoary vine
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Sorry

worldly linden
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u get that alot im assuming

hoary vine
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Some people kept spamming questions the whole day.

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After I added them

worldly linden
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yeah i get that

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okay well hopefully if i have anymore questions u will see them haha

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thanks again

hoary vine
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You're welcome. I will be more likely to answer questions of people I like and that I know are trying.

worldly linden
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well i will definitely be trying haha

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super enjoyable subject

hoary vine
worldly linden
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okay will do

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one last thing

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are u a math teacher or something?

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or student?

hoary vine
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Nope, just studying maths myself

worldly linden
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nice one

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at uni or on ur own/

hoary vine
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Hoping to be allowed to tutor in a year or so

hoary vine
hoary vine
worldly linden
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u already took real analysis im guesing?

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and yes I am

hoary vine
worldly linden
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is that a 100?

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or something

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im in US

hoary vine
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Like perfect

worldly linden
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good shit

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did u study alot or just comes to u easy?

hoary vine
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I remember this one from my preparation.

hoary vine
worldly linden
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well that is reassuring

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I am definetly having to work at this

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but it is getting easier as I become more familiar

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any tips when writing proofs?

hoary vine
hoary vine
worldly linden
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ur very detailed so i admire that