The following is an excerpt from Chapter 10 of Calculus, Early Transcendentals by James Stewart. It has been reformatted slightly from its appearance in the textbook so that it corresponds to the LaTeX article style.
1 The Binomial Series
You may be acquainted with the Binomial Theorem, which states that if +, and are any real numbers and 5 is a positive integer, then 5Ð5 "Ñ 5Ð5 "ÑÐ5 #Ñ Ð+,Ñ5 + 5 5+ 5" , + 5# , # + 5$ , $ #x $x 5Ð5 "ÑÐ5 #ÑâÐ5 8"Ñ â +58 , 8 8x â5+,5" , 5
The traditional notation for the binomial coefficient is 5 5 5Ð5 "ÑÐ5 #ÑâÐ5 8"Ñ " 8"ß#ßáß5 !8 8x which enables us to write the Binomial Theorem in the abbreviated form 5 5 Ð+ ,Ñ5588 " + , 8 8!
In particular, if we put +" and ,B , we get 5 5 Ð"BÑ58 " B Ð Ñ 8 1 8!
One of Newton's accomplishments was to extend the Binomial Theorem (Equation 1) to the case where 5 is no longer a positive integer. In this case the expression for Ð"BÑ5 is no longer a finite sum; it becomes an infinite series. Assuming that Ð"BÑ5 can be expanded as a power series, we compute its Maclaurin series in the usual way: 0ÐBÑÐ"BÑ5 0Ð!Ñ" 0ÐBÑ5Ð"BÑw5"w 0Ð!Ñ5 0ww ÐBÑ 5Ð5 "ÑÐ" BÑ 5# 0 ww Ð!Ñ 5Ð5 "Ñ 0www ÐBÑ 5Ð5 "ÑÐ5 #ÑÐ" BÑ 5$ 0 www Ð!Ñ 5Ð5 "ÑÐ5 #Ñ ãã 0Ð8Ñ ÐBÑ 5Ð5 "ÑâÐ5 8 "ÑÐ" BÑ58 0 Ð8Ñ Ð!Ñ 5Ð5 "ÑâÐ5 8"Ñ
1 __0Ð8Ñ Ð!Ñ 5Ð5 "ÑâÐ5 8"Ñ Ð"BÑ58 "" B B 8 8x 8x 8! 8!
If ?88 is the th term of this series, then ? 5Ð5 "ÑâÐ5 8 "ÑÐ5 8ÑB8" 8x 8" ººº8 º ?8 Ð8 "Ñx 5Ð5 "ÑâÐ5 8"ÑB kk58 ¸¸" 5 lBl 8 lBl Ä lBl 8 Ä _ 8" " as " 8 Thus, by the Ratio Test, the binomial series converges if lBl" and diverges if lBl" .
Definition 1 (The Binomial Series ). If 5 is any real number and lBl" , then 5Ð5 "Ñ 5Ð5 "ÑÐ5 #Ñ Ð"BÑ5#$ "5B B B â #x $x _ 5 B" 8 8 8! where 5 5Ð5 "ÑâÐ5 8"Ñ 5 Ð8 "Ñ " 88x!
We have proved Definition 1 under the assumption that Ð"BÑ5 has a power series expansion. For a proof without that assumption see Exercise 21. Although the binomial series always converges when lBl" , the question of whether or not it converges at the endpoints, " , depends on the value of 5 . It turns out that the series converges at 1 if "5! and at both endpoints if 5 ! . Notice that if 5 is a 5 positive integer and 85 , then the expression for Ð8 Ñ contains a factor Ð55Ñ , so 5 Ð8 Ñ! for 85 . This means that the series terminates and reduces to the ordinary Binomial Theorem (Equation 1) when 5 is a positive integer. Although, as we have seen, the binomial series is just a special case of the Maclaurin series, it occurs frequently and so it is worth remembering.
" Example 1. Expand Ð"BÑ# as a power series.
Solution. We use the binomial series with 5# . The binomial coefficient is # Ð#ÑÐ$ÑÐ%ÑâÐ#8"Ñ 88x Ð"Ñ8#$%â8Ð8"Ñ Ð"Ñ8 Ð8 "Ñ 8x
2 and so, when lBl" , "#_ Ð"BÑ# " B 8 Ð"BÑ# 8 8! _ " Ð"Ñ88 Ð8 "ÑB ¨ 8!
Example 2. Find the Maclaurin series for the function 0ÐBÑ"ÎÈ %B and its radius of convergence.
Solution. As given, 0ÐBÑ is not quite of the form Ð" BÑ5 so we rewrite it as follows: "" ""B"Î# "B ÈÈB %B É#"% # % %"%
" Using the binomial series with 5 # and with B replaced by BÎ% , we have ""B"Î# "B_ " 8 " "# È #% # 8 % %B 8! ""B"$ B#$ "$& B "## ### # # % #x % $x %
"$& â " 8" B8 â ### # â 8x % " " "$ "$& "$&âÐ#8"Ñ " B B#$ B â B 8 â # ) #x )#$ $x ) 8x ) 8
We know from Definition 1 that this series converges when lBÎ%l " , that is, lBl % , so the radius of convergence is V% . ¨
3