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Review of Power Series MAT 201E
2023/ Astronautics
One of the simplest power series may be given as a polynomial function that is the sum
of a finite number of terms.
ଵ
ଶ
ଶ
Also, some elementary functions can be expressed as infinite power series, such as,
௫
ଶ
ଷ
ஶ
ୀ
cos 𝑥 = 1 −
ଶ
ସ
ଶ
ஶ
ୀ
sin 𝑥 = 𝑥 −
ଷ
ହ
ଶାଵ
ஶ
ୀ
ଶ
ஶ
ୀ
In short, a series equal to the sum of infinite number of the powers of x (𝑖. 𝑒. 𝑥
) that
are multiplied by respective constants (𝑖. 𝑒. 𝑎
ଵ
ଶ
ଶ
ஶ
ୀ
is called a power series. This form of the power series may be expressed in a more
generalised way by shifting the point 𝑥
to the origin, as follows,
ଵ
ଶ
ଶ
ஶ
ୀ
Review of Power Series MAT 201E
2023/ Astronautics
SOME USEFUL NOTES
- A power series
ஶ
ୀ
is said to converge at a point x if
lim
→ஶ
ୀ
exist for that x. The series certainly converges for 𝑥 = 𝑥
; it may converge for all x, or it
may converge for some of values of x and not for others.
- The series
ஶ
ୀ
is said to converge absolutely at a point x if the series
ஶ
ୀ
ஶ
ୀ
converges. It can be shown that if the series converges absolutely, then the series also
converges; however, the converse is not necessarily true.
- One of the most useful tests for the absolute convergence of a power series is the
ratio test. If 𝑎
≠ 0 , and if, for a fixed value of x,
lim
→ஶ
ାଵ
ାଵ
lim
→ஶ
ାଵ
then the power series converge absolutely at that value of x if
𝐿 < 1 and
diverges if
𝐿 > 1. If
𝐿 = 1 , the test is inconclusive.
- If the power series
ஶ
ୀ
converges at 𝑥 = 𝑥 ଵ
, it converges absolutely for |𝑥 − 𝑥
ଵ
|; and if it diverges at
ଵ
, it diverges for |𝑥 − 𝑥
ଵ
Review of Power Series MAT 201E
2023/ Astronautics
- The function 𝑓 is continuous and has derivatives of all orders for
Further, 𝑓′, 𝑓′′,… can be computed by differentiating the series termwise; that is,
ᇱ
ஶ
ୀ
ିଵ
ஶ
ୀଵ
ᇱ
ஶ
ୀ
ିଶ
ஶ
ୀଶ
()
ି
ஶ
ୀ
Each of the series converges absolutely for
< 𝑅. So, for 𝑥 = 𝑥
, each differential
can be found as follows,
ᇱ
ଵ
ᇱᇱ
ଶ
()
Rearranging the general equation (i.e. the last equation in the line above), we get
()
- If the function 𝑓 can be expressed as
()
ஶ
ୀ
about 𝑥 = 𝑥
, then this series is called Taylor series
1
for the function f about 𝑥 = 𝑥
- A function f that has Taylor series expansion about 𝑥 = 𝑥
()
ஶ
ୀ
with a radius of convergence 𝑅 > 0, is said to be analytic at 𝑥 = 𝑥
. For example, sin 𝑥
and 𝑒
௫
is analytic everywhere, 1/𝑥 is analytic except at 𝑥 = 0, and tan 𝑥 is analytic
except at odd multiples of 𝜋/2. According to the Notes #6 and #7, if 𝑓 and 𝑔 are analytic
at 𝑥
, then 𝑓 ± 𝑔, 𝑓 ∙ 𝑔 and 𝑓/𝑔 (provided that 𝑔(𝑥) ≠ 0 ) are also analytic at 𝑥 = 𝑥
1
Named after English mathematician Brook Taylor (1685-1731).
Review of Power Series MAT 201E
2023/ Astronautics
- If
ஶ
ୀ
ஶ
ୀ
for each x in some open interval with centre 𝑥
, then 𝑎
for 𝑛 = 1,2,3 …. In
particular, if
ஶ
ୀ
for each such x, then 𝑎
ଵ
ଶ
References:
Boyce, W. E.; DiPrima R.C.; “Elementary differential equations and boundary
value problems”, Eight Edition, John Wiley & Sons Inc, 2005.
Cesur, Y. “Diferansiyel denklemler ve Mathematica”, Sixth Edition, 2017.
