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# Worked example: divergent geometric series

AP.CALC:
LIM‑7 (EU)
,
LIM‑7.A (LO)
,
LIM‑7.A.3 (EK)
,
LIM‑7.A.4 (EK)

## Video transcript

so we've got this infinite series here and let's see it looks like a geometric series when you go from this first term to the second term we are multiplying by negative 3 and then to go to the next term we're going to multiply by negative 3 again so it looks like we have a common ratio of negative 3 so we could actually rewrite this series as being equal to negative 0.5 I could say times negative 3 to the 0 power negative 3 to the 0 power plus plus negative 0 or maybe I could just keep writing this way minus 0.5 times negative 3 to the first power times negative 3 to the first power minus 0.5 minus 0.5 times negative 3 to the second power and they get a 3 to the second power and we're just going to keep going like that and we could just say we're just going to keep having minus 0.5 times negative 3 to eat or to higher and higher and higher powers or we could write this in Sigma notation this is equal to the same thing as the sum from let's say N equals 0 to infinity we're just going to keep going on and on forever and it's going to be this first it's going to be you could kind of think the thing we're multiplying by negative 3 to some power so it's going to be negative 0.5 actually must do that yellow color so it's going to be negative 0.5 times negative 3 negative blue color so times negative 3 to the nth power here this is what n is 0 here is n is 1 here is n is equal to 2 so we've been able to rewrite this in different ways but let's actually see if we can evaluate this and so we have a common ratio of negative 3 so our R here is negative 3 and the first thing that you should think about is well in order for this to converge our common ratio the the magnitude of the common ratio or the absolute value the common ratio needs to be less than one for convergence convergence and what is the absolute value of negative three well the absolute value of negative three is equal to three which is definitely not less than one so this thing will not converge this thing will not converge and even if you look at this it makes sense because the magnitudes of each of these terms are getting larger and larger and larger we're flipping between adding and subtracting but we're adding and subtracting larger and larger and larger and larger and larger values intuitively when things converge you're kind of that the each successive term is get it tends to get diminishing ly small or maybe it cancels out in some type of an interesting way but because the absolute value of the of the common ratio is is greater than or equal to one in this situation this is not going to converge to a value