MA 16600 Extra Credit Due the date of your final exam. Name (+10i pts ) These bonus points will be added to your Final Exam score. How Euler Astonished the World with the Amazing Fact elz = cos z + i sin z and Other Amazing Facts On the Journey to It Once upon a time (1748) there was a mathematician named Euler, who at the age of 35 used infinite series to find some amazing facts. The goal of this activity is to show how he arrived at the Amazing Fact in the title of this activity, but more beautiful gems will be shown along the way. In particular, ... Amazing fact #1 The infinite series O! 1 ! 2 ! 3 ! + ... = e= 2.71828 1828 459045... The series 1 1 1 1 0! '1! 2! 3! + ... converges quickly. We can write a sequence of partial sums Si, S2, S3, ... SI = 1 NORMAL FLOAT AUTO a+bi RADIAN MP S2 = 1 + 1 =2 - + 1 + 1 - = 2.5 Plot1 Plot2 Plot3 0! 1! 1 12! Y18X+1 1 1 1 1 S4 = 8 + 0! 1! 2! + = 2.6 3! 1 1 1 1 1 Ss Inth Partial Sum! + + + = 2.7083 0! 1! 2! 3 ! 41 24 Denominator Term #, n Sn X Y1 Y 2 S6 = + 1 1 1 1 163 + = 2.716 1 ! 2! 3! 4 ! 5 ! 60 N PO S7 1 1 1 1 1 1957 3 + + 2! + O! 1 ! 3 ! 4! 5 ! 6 ! = 2.71805 720 3 4 S8 = 1 1 1 1 + 1 1 685 + 2 ! 3 ! 4! 5! 6! 7! 252 ~ 2.71825 4 5 65 24 5 6 163 60 So = - 1 1 1 1 1 1 1 1 1 109601 7 1957 O! 1! 2! 3! 4! 5! 6! '7! 8! 40320 = 2.7182787698413 720 685 252 2.7183 (i) 1. For what value of n does Sn first exceed 2.71828 1828 459? n= Sn Yz=2. 7182787698413 e= 2.71828 1828 4590 45... 25 1.5- 0.5 8 nAmazing fact #2 If w is any complex number, the infinite series 1 + w 1! + - 2! 4! + ... = Notice this equation is true for w = 0 since e" = 1 and 1 + 0+0+0+ ...=1. Amazing fact #9 Notice this holds for w = 1 by Amazing Fact #1. 6. Find the val Comple Amazing Fact #3 Pow (1) 2. Suppose w = -1 in Amazing Fact #2. Write the first thirteen terms of the infinite series expansion of c raise. to the power of -1. Powers of i should be simplified. Double check you only report thirteen terms. Amazing Fact #4 If w = i in Amazing Fact #2, then e' = 2 ! 3 ! 4 ! + ;10 5 ! 7 ! 8 ! 9 ! We can simplify this just a little using powers of i, where i = -1, i' =-i, etc. 10! 1 1! + 12 ! + ...... Rewrite the expansion with simplified numerators of ti or 1 1, by carefully completing the boxes below. e' = Ly i 1 1 2 ! 3 ! 4 ! 5! 6! 7 ! + 8! 9! (i) 4. Suppose w = iz in Amazing Fact #2, where z is any complex number. 10! 11! + 12! Write the first thirteen terms of the infinite series expansion of e raised to the power of iz. Powers of i should be simplified. Double check you only report thirteen terms. 12 Amazing fact #5 If w is any complex number in radians, the infinite series 1 w 2 + wo w 8 2 ! wi W/ 14 4 ! 6 ! 8 ! 10! 12! 14! + ... = COS w Notice this is true for w = 0 since cos 0 = 1 - 0+0- 0+ ... = 1 Amazing fact #6 If w is any complex number in radians, the infinite series w + w' w' will w13 wis 1 1 3 ! 5 ! 7 ! 9 1 11 ! + 13! 15! + ... = sin w Notice this is true for w = 0 since sin 0 = 0-0 +0-0+ ... =0 Amazing facts #7 and #8 (27) 5. Complete the boxes with simplified numerators of ti or + 1 to find the value of sin i and cos i. Powers of i should be simplified cosi = + ...... 0! 2! 61 10! sin i = f .... ... 1! 3! 7! 9! 171 3! 5! 7! 91 1 1!Amazing fact #9 (1 ) 6. Find the value of cos i + isin i and write it as a power of e. Complete the boxes with simplified numerators of ti or + 1. Powers of i should be simplified. Use Amazing Facts #3, #7, and #8. cosi + i sini = + 21 . ... 41 81 101 1! 31 +... 1! 0! 11 21 31 41 6! 7! 8! 9! 10! (i) 7. From Amazing Fact #2, we found the value of e raised to the power of iz, where z is any complex number. Now write the sum in two rows of terms, separating even and odd powers. Complete the boxes with simplified numerators of ti or + 1: Powers of i should be simplified. + . 28 1-210 8! 10! + 3 ! 5! + 71 91 TIP: Check that if you substitute z = 1 you get Amazing Fact #1. (.57) 8. What does the first row of terms of even powers equal? (Your expression involves z.) (.51) 9. What does the second row of terms of odd powers equal? (Your expression involves z.) Amazing Fact #10 (i) 10. Use your answer to the previous two questions to write e in terms of the trig functions. Hint: Recall the goal of this activity