- Write a formula for the nth term of the given infinite sequence:
- Sketch a rough graph of the sequence.
- Take the problem steps by steps. Analysis the sequence and find what you know and can solve. For example, the exponents on e, the exponent is 12. 22, 32, 42... By knowing this we know that e^n2, gives the answers. So next to do if find out how to create the alteration of positive and negavite sings. (-1)n+1.
The answer Tn=(-1)n+1e^n2.
lets check:
T1=(-1)1+1(e1^2)
T1=(-1)2 (e1)
T1=e
Lim= DNE (each time the points on the graph are further apart.)
Sybil the Rat analysis:
S is a geometric series with r=1/2
Sn= distance covered= (1/2) 1-(1/2)^n/ 1-1/2
Since it is geometric we take the Lim of each side. Remember lim must be 1 in order for the rat to cross the room. (as n approches infinity.)
LimSn= Lim(1/2) 1-(1/2)^n/ 1-1/2
LimSn= [1-(1/2)^n]
LimSn= Lim(1-0)
LimSn=1
Tortoise and achilles Geometric series.
Asume that the tortoise gets a 10-meter head start and that Achilles runs 10 times as fast. At what distance and at what time will achilles catch the tortoise?
Tn= distance that achilles hast to run.)
first few terms: 10+ 1 + 1/10+ 1/100..+Sn..
r=1/10; T1=10
Sn=T1[1-(1/10)^n/(1-1/2)]
take the lim of each side, asume that n is approching infinity.
LimSn=lim[1-(1/10)^n/(1-1/10)]
¨(1/10)^n¨- based on the book theorem: If /r/less than 1, the Limr^n=o. as n approches infinity.
LimSn=lim[1/(1-1/10)]
LimSn= (10/9)/10
LimSn=100/9 = 11.1111 meters achilles will catch the tortoise.
Notice that the n in both problem the formula simplifies to:
Sn=T1/1-r
Theorem: Sum of an infinite geometric series:
If /r/ less than infinite geometric series converges to the sum.
Sn=T1/1-r ; T1+T2+T3+T4+...Tn
If /r/greater or equal to 1, and T1 is not equal to 0, then the series diverges. meaning that every time the points are getting further apart.
(Tortoise and achilles problem can be done with 2 linear equations.)
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