Alright so in This Video I'm going to do some Series examples
Basically These Are all going to Apply to Geometric Series so again the Question is going to be is it converge
Or diverge
and
If it Converges
We'll Say to what Value?
so suppose my first Series Is
1 plus
0.4 plus 0.16
Plus Point Zero Six Four We'll Assume This Pattern Continues
Notice the Number I'm Multiplying by it's Pretty Pretty Easy to spot on the first one it looks like I'm Multiplying by 0.4 and
Notice if I multiply 0.4 by 0.4 again I'll get 0.16 [I'm] Multiplying
0.16 By 0.4 I'm going to get Point Zero Six Four so this Point Four is going to be my Common ratio R
and
If I wanted to, well let's just stop There There's no Point Really and Rewriting it with a summation it
Says This Is going to converge Because remember it Says if this number R
It Says if R Is between Negative 1 and Positive 1 it does Converge
Well Point [4] is Certainly in This interval and it's going to converge to the first Term
Well Here's The first Term that Gets spit spit out?
1 over 1 Minus 0.4 Remember You Take the first Term over Always 1 Minus the Common Ratio
Well This Is 1 over 0.6?
1 Minus 0.4 0.6. I can write Point 6 IS 6 Tenths
That's 10 6 if I flip it and that's equivalent to
5/3 Which Is 1 and 2/3 Which IS 1.6
Repeating
So the idea is if I Keep Adding up these Numbers
Infinitely Many of Them I'm going to get as close as I want to to the [Number] 1 Point 6
Repeating so yes This is A convergent gee Metric Series
Let's Look at [Another] one Here
Suppose I have let's Look at
Let's Just Make One [Up] suppose I have n equals
Three to Infinity
And Let's Make it five Times [Two] Thirds Say to the how about
in Minus 1
okay, so
Again I look at this and I think [okay] [2/3] I've got a number Raised to a variable Power
This Clues Me in That it's a geometric Series
Again This Number 2/3 Is between Positive 1 and Negative 1 so that means it is going to converge
and
If you Just Automatically Set, oh 5s my first Term over
1 Minus 2/3 that's Actually going to be incorrect in This Problem, [okay] that's why I think Sometimes When They write?
This A over 1 Minus R
Stuff
They'll Usually Write the Geometric Series
Starting At N Equals 1 and Notice in This case my n IS 3
Well what is the first Term?
The First Term is Simply Going to be the [first] Number [that] Gets spit out when you plug in the first
Number Where your Series Starts
So in This case if I plug in n Equals 3 I'm going to get 5 Times
2/3 Raised to The 3 Minus 1
Well 3 Minus 1
That's 2/3
Squared
2/3 Squared That'S 4/9
So I'll get 5 Times 4/9
I'll get 20 over 9 and This is now my first Term of this Series
So this Series Again Definitely Does Converge but it's not going to converge to this Number
It's going to converge to 20 Divided by 9
Over 1 Minus the Common Ratio
2/3 and I'll Let you Simplify that one Down. [I] don't feel like doing that one
But this is the basic Idea to get the first Term
Simply Plug in Wherever the Index is starting Into your Formula and
Whatever You get out, well that's your first Term?
Clearly if I started this at 1 Or
You are at a Billion I'm going to get A much Different first Term in all of Those Cases
So that's the Idea on how to get your first Term so again just be Careful
This Could, be you know
20 in Well, We don't want to Use 20 in
Maybe I've got in Just [Plano] in to the [empower]
Again then my first Term Is going to be a little bit Different if I don't have the Minus One
I'll get five Times TWo-Thirds
Cubed and That's going to give me A very Different first Term Than [Twenty], over nine so just be Careful about that
Let's See, Let's find Another example Here
So again Suppose, We want to know Does it converge or diverge?
Suppose I have n Equals one To [Infinity]
Suppose I have [Pi] Raised to the n
Over 3
Raised to the N [plus] 2 Power
Well in This case i'm going to try to Make it Look more like a geometric Series so I want a single Number
Raised To a Power
I'm going [to] write so i've Still got. Pi to, the n on Top I'm going to write this as 3 to the n
Times 3 Squared, ok, Right again, I've got like Bases if I add up the exponents I'll get N plus 2 I
Can Pull The [3] Squared out Front The 1 over 3 [Squared] that will Give me
1/9
So i'm getting Rid of The 3 Squared here [and] i'm Left With the Series from n Equals 1 to Infinity and
I'Ve Pulled out The 3 Squared [I've] Now got
[Pi] to the n over 3 to the [n] so i can [write] [that] as pi over 3 to
the N
So again I now See a number Raised to a variable Power
Does this Converge Or Diverge
Well
We have to think About our Common Ratio?
Definitely Pi Over 3 is bigger Than 1 and
Remember our Common Ratio inside of here has to be between Negative [1] and Positive [1] to Converge
Well Since This number falls Outside of that interval, we Can Conclude that this is A
Divergent
Geometric Series
ok, so Obviously
[We] don't have to do the first Term over 1 Minus the Common Ratio Because it's Simply Simply not Going to converge at all
Let's do Two more Here
Suppose I have the Series N Equals 1 to Infinity
[of] Let's Say N Squared, over
2 [and] Squared Minus 1
So the first thing When I see this Is this looks like almost to me like a limit Problem I'm so used to seeing Those
Rational Functions Polynomial on Top and Polynomial on the Bottom and Notice if You Take the limit of This
The Limit as N goes to Infinity of N Squared
Over 2 [and] Squared Minus 1
Well This limit you Can either Divide Everything Through [by] n Squared?
[There's] That Little Trick if The highest Power on Top is the same as the highest power on the Bottom
We can Just Take the ratio of Their coefficients and That will be my answer
Well Since The limit of This Does not equal Zero We Can Conclude that this original Series Diverges?
By the Test for Divergence
and
Again we're Finished so
Think About This Little Trick It'll Make so Many Problems [so] [much] Easier for you again if You're a little Shaky with limits
My Advice Would be [to] go back and Review Limits
You've Had to [have] [Seen] limits before you've Done Sequences in Series Series Are One of Those things that are Really
Tricky Limits Can Be Tricky as, well but you've Certainly Seen that stuff at least and
I think with a little Practice if You have Gotten Rest you on your limits you Can bring A lot of it back Pretty Quickly
Let's do One More example here for the road
Suppose I have
This Series
3 to the [N]
Plus 2 to the N, over
6 to the N in
This Example I'm Just going to, Bust up my Fraction