 what is what is the formula we arrived at we arrived at sum of n terms of a gp in n is equal to a r to power n minus one okay let's say modulus of r is greater than one right what does this mean this means minus one uh r is less than minus one or r is greater than one in both these cases this will be satisfied here modulus of r mod r absolute value of r is greater than one here below then tends to infinity that means you have very large series lots of terms don't you think r to the power n will tend towards infinity if r is greater than one or less than minus one so minus infinity plus infinity whichever way the magnitude mod of r n will tend towards infinity yes or no if r is greater than one let's say 3.5 and you are raising it to power of 10 to the power 6 isn't it a huge number very big number so that means so if you have an infinite series what is an infinite series it means when n tends to infinity very large number of some terms large number of terms right with mod r greater than one okay with mod r greater than one then mod of r n will tend to infinity again is this statement clear clear clear to everyone is this statement clear for an infinite series that means what is an infinite series n tends to infinity number of terms is huge and if the common ratio is greater than one or less than minus one then the absolute value of r to the power n will tend towards infinity yes or no right so in infinite series greater than r is greater than one this form is infinite negative positive whichever way it is but infinite infinite sum clear is it okay but same infinite series that means n tends to infinity but mod r now is less than one what do you think r to the power n will be tending to if now r is very less small amount yes so it will tend to 0 why because let's say r is equal to 0.5 right so as you increase the power of 0.5 r r square will be 0.25 it is going down see then r cube is 0.125 going down further correct r to the power 4 is 0.0625 going further down right r to the power 5 is 0.03125 it is going down and down and down and down and down right so if r to the power very large value of n it will always be near 0 clear yes so yeah so why do you want to express it like that okay what is the point okay so understand the importance of this so is this clear whichever way you want to express but the final thing is r to the power n is going towards 0 is r equals to 0 actual or is it simplified r is 0 r can never be 0 as it can mute and say I can't understand r is not 0 first of all we are not taking r as 0 r cannot be 0 anyways you just learned in a GP r cannot be 0 then we can't define a GP here we are not talking about r to be 0 we are saying if r is very small then r to the power n will be tending towards 0 it's not equal to 0 tending towards and very very very very small amount of yeah it's a very small number okay right so hence I have not used this expression I am not saying this no I am saying as n tends to infinity that means n is becoming bigger and bigger r n becoming 0 okay right now tell me r n can it be can it ever breach the limit of 0 can it be you know can it become minus 0.01 ever however much n is is this possible is this possible this will never be possible that means what limiting value that is the boundary boundary line limiting value of r to the power n as n tends to infinity is 0 so r to the power n is not 0 its limit is 0 do you get the difference it will always be closer to 0 but it will never breach the value of 0 do you get the point right so we are saying we are not saying r n is 0 we are saying r n is limiting limiting value of r n is 0 as n tends to infinity now again come back to the same equation s n is equal to a okay now condition is mod r less than 1 now why am I using mod r please understand this also so even it if it is minus 0.5 let's say raise it to 1000 don't you think it will start going towards 0 sir irrespective of right so if r is positive let's say this is the number line right if r is positive r to the power n goes towards 0 like this if r is negative then r to the power n goes towards 0 like that clear right the value as you keep on increasing the power the value keeps on sifting like this and if I if you increase it here it keeps on sifting like that so both ways it is going towards 0 is it okay now if you understand that part then go below so what do we understand then there is a fixed value of r n which it cannot breach what is that value 0 correct so I am writing limit n tends to infinity s n will be how much as n tends to infinity what will be the limiting value of s n now some is obviously r n has a limiting value of 0 so you put a 0 minus 1 divided by r minus 1 yes or no so this will be simply a 1 minus r do you get that right this is a very interesting phenomena in maths very very interesting why it is interesting let's understand this by an example so let's say you are adding this s n 1 plus half plus 1 by 4 plus 1 by 8 plus 1 by 16 plus 1 by 32 so on so forth yes infinite series right so now this is going towards infinity let's say infinite terms are there correct so the interesting fact is however be the number of term this s n converges the word is converges so let's try few and then and tell you how what does it mean converges what will be s 1 below s 1 is how much s 1 is how much tell me s 1 value some of yes some of the series with only one term is one very good what is s 2 s 2 is 1.5 very good what is s 3 s 3 is 1.75 isn't it right what is s 4 1.875 am i right am i right s 4 is this much 1.875 only yep and then keep on you're adding this keep on adding this as you move forward you will see if you find out till 100 now see the sum here let me okay so this is the limit limitation of excel so let's say still it is not 2 but you can see as you are adding all these but so let me increase the font size can you see that this value is it is it visible there correct this is 1.99999 so you keep on adding if you keep on adding let's say if i add also some more terms okay and here also some more terms now let's see what what happens to the sum sum may be 100 here it's coming out to be 2 correct and what if if i change 2 to 3 if i change 3 see it becomes 1.5 see if i if i keep 3 that is what is this the the sum is 1 plus 1 by 3 plus 1 by 9 plus 1 by 27 plus 1 by 81 till infinite terms what will this value be 1.5 right it is converging converging to this as you are increasing the number so if you see now now look at this position here can you see this this point all of you some can you see this thing guys here so just confirm if you can see the sum here it's some sum is mentioned here can you see that just keep watching this one so as i will move down see the sum which is varying 1.4938 and now see it's it's now you know nothing is happening to 4 and 9 only the decimal places far away from the decimal point is getting impacted getting it see it will never breach the value of 1.5 correct what if i do it as 4 see 1.33 so that means if you have this 1 plus 1 by 4 plus 1 by 16 plus 1 by 64 plus 1 by 256 so on and so forth if you take it to infinite value you'll get what 1.33 okay and what if it is 5 see 1.25 whatever what if it is 6 so here also you will reach up can you tell me till what value what will be the last value of this one if i take 100 here see what will happen can you see that can you can you tell me where will it end what is the limiting value of the sum itself so 1000 if you take see can you see that if you take now it cannot see up to this number it is coming out to be 1.000 so hence this limiting sum also limiting value is 1 correct you will never get if you are starting with one all obviously if you're starting with one plus something the final value if you have one plus let's say one by one million one millennium oh one one million plus one by this plus one by this and what will happen till infinity what was this sum this whole sum is tending towards one is that okay here interesting this is what is called converging series converging series clear so all what is the now what is the condition for converging series so condition for converging series is so what is the converse converging series now you've got this sum so i'm now writing s infinity why because infinite terms are there and what was the formula we had arrived at arrived at one a minus a upon one minus r this is the sum of a converging series converging series right with modulus of r less than one i hope this is clear to everyone is this clear crystal clear yes or no so we learned what is the summary of the previous discussion summary of previous discussion is sn so two options finite if it is a finite series whatever be the value of r the sn is a r to the power n minus one upon r minus one correct if infinite right then again two cases if mod r is less greater than one then sn will either be infinity or negative infinity whichever right sn itself will become infinite but if mod r is less than one then sn or s infinity is a upon one minus