Video Transcript
Is the infinite geometric series 10
add six add 18 over five add 54 over 25 add 162 over 25 and so on convergent?
We know that a series can either
converge or diverge. A convergent series is a series
which has a finite limit. Otherwise, we have a divergent
series. In order for an infinite geometric
series to converge, weโre going to need the terms to get smaller and smaller. Otherwise, we find that the terms
get bigger and bigger, and the sum of all the terms diverges to โ. It does appear here that the terms
of the sequence are getting smaller, so weโre tempted to say that weโre able to find
the sum to be a finite number. But there is a way that we can
check.
For an infinite geometric series ๐
one add ๐ one ๐ add ๐ one ๐ squared and so on where ๐ one is the first term and
๐ is the common ratio, so this means the number we multiply to get to the next
term, as long as this number is between negative one and one, itโs small enough for
the terms of the sequence to be getting smaller and smaller and therefore
converge. In other words, an infinite
geometric series converges as long as the absolute value of ๐ is less than one. This just means that ๐ is strictly
between negative one and one. But what is ๐ for this series?
Well, because we multiply each term
by ๐ to get the next term โ for example, we multiply the first term by ๐ to get
the second term โ then if we take the second term and divide it by the first term,
we find ๐. So for this series, ๐ equals the
second term, which is six, divided by the first term, which is 10. And in fact, for this, we could use
any two consecutive terms, but weโll use the first and the second term seen as
theyโre easier numbers to work with. So we have that ๐ is equals six
over 10 which is, equivalently, three-fifths.
And remember that we said that an
infinite geometric series converges if this common ratio ๐ is between negative one
and one. And because three over five lies
between negative one and one, we can say yes; this infinite geometric series does
converge.
