Understanding Geometric Series with Complex Numbers
Learn how to apply geometric series concepts to complex numbers, including using the exponential form for summation and spotting patterns to simplify calculations. Explore Euler's identity to deepen your understanding and discover the beauty of mathematical relationships in this engaging educational content.
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Teachings for Teachings for Exercise 1E Exercise 1E
Complex Numbers You can use the results for Geometric series with complex numbers ? = the first term ? = the common ratio ? = the number of terms Sum to infinity of a geometric series, valid when ? < 1 ??=? 1 ?? ? Sum of a geometric series ? = 1 ? 1 ? For ?,? The alternative form is also ok! ? ?? 1 ? 1 ? = the first term ? = the common ratio ? = the number of terms Sum of a geometric series with complex numbers =? 1 ?? 1 ? Sum to infinity of a geometric series with complex numbers, valid when ? < 1 ? = 1 ? 1E
? 1 ???= ? + ?? + ??2+..+??? 1=? 1 ?? ? ???= ? + ?? + ??2+..= 1 ?, ? < 1 1 ? ?=0 ?=0 Complex Numbers You can use the results for Geometric series with complex numbers Formally, it looks like this For ?,? ? 1 = ? + ?? + ??2+..+??? 1=? 1 ?? ??? 1 ? ?=0 ? ???= ? + ?? + ??2+.. = , ? < 1 1 ? ?=0 1E
? 1 ???= ? + ?? + ??2+..+??? 1=? 1 ?? ? ???= ? + ?? + ??2+..= 1 ?, ? < 1 1 ? ?=0 ?=0 1 2 1 2? Complex Numbers ????+ ? ???= ????? ???? ? ???= ????? ??+1 ??= 2????? You can use the results for Geometric series with complex numbers Use the exponential form, ? = ??? 1 ????+ ??? ?= 2????? Rewrite You will also need a variation on a result from the previous section (you should hopefully have encountered this in exercise 1D!) ????+ ? ???= 2????? Divide by 2 1 2 ????+ ? ???= ????? ??+1 ?? 1 ??= 2????? ??= 2?????? ?? 1 ??= 2?????? Use the exponential form, ? = ??? 1 You will need to learn to spot situations where you can create this pattern in order to use it! ???? ??? ?= 2?????? Rewrite ???? ? ???= 2?????? Divide by 2i 1 2? ???? ? ???= ????? 1E
? 1 ???= ? + ?? + ??2+..+??? 1=? 1 ?? ? ???= ? + ?? + ??2+..= 1 ?, ? < 1 ???= 1 1 ? ?=0 ?=0 1 2 1 2? Complex Numbers ????+ ? ???= ????? ???? ? ???= ????? ? 1 ?? 1 ? You can use the results for Geometric series with complex numbers ?? ? In this question, ? = 1, ? = ? and ? = ? ? ?? ? =1 ? 1 ? ?? ? ? ?+ ???? ? ?, Given that ? = ??? where ? is a positive integer, show that: We can simplify the numerator =1 ??? 1 ? ?? ? ? 2? 1 + ? + ?2+..+?? 1= 1 + ???? Start with Euler s relation You will need to use the exponential form of ? in order to then use the patterns we just saw ???= ???? + ????? Let ? = ? ???? + ????? = ??? ???= ???? + ????? Calculate terms on RHS ? ? ? ? = ??? ???= 1 + ?(0) ??? + ???? ? Simplify ?? ? ???= 1 = ? Add 1 (don t confuse ?symbols with the letter n !) ???+ 1 = 0 This is known as Euler s identity , and is one of the most famous relationships in all of Maths! 1E
? 1 ???= ? + ?? + ??2+..+??? 1=? 1 ?? ? ???= ? + ?? + ??2+..= 1 ?, ? < 1 ???= 1 1 ? ?=0 ?=0 1 2 1 2? Complex Numbers ????+ ? ???= ????? ???? ? ???= ????? ? 1 ?? 1 ? You can use the results for Geometric series with complex numbers ?? ? In this question, ? = 1, ? = ? and ? = ? ? ?? ? =1 ? 1 ? ?? ? ? ?+ ???? ? ?, Given that ? = ??? where ? is a positive integer, show that: We can simplify the numerator =1 ??? 1 ? ?? ? ? 2? The numerator will therefore be equal to 2 Now we need to manipulate the fraction to create one of the patterns above. The denominator has a subtraction, so we are aiming for that one Multiply all terms by ? 1 + ? + ?2+..+?? 1= 1 + ???? 2 = You will need to use the exponential form of ? in order to then use the patterns we just saw ?? ? 1 ? ?? 2? 2? ?? 2? ? ?? 2? ???? + ????? = ??? = ?? 2? ? Now multiply all terms by -1 (the terms on the denominator have had their positions switched) ? ? ? ? = ??? ??? + ???? ? ?? 2? 2? ?? 2? ? ?? ? = = ? ?? 2? ? (don t confuse ?symbols with the letter n !) 1E
? 1 ???= ? + ?? + ??2+..+??? 1=? 1 ?? ? ???= ? + ?? + ??2+..= 1 ?, ? < 1 ???= 1 1 ? ?=0 ?=0 1 2 1 2? Complex Numbers ????+ ? ???= ????? ???? ? ???= ????? ?? 2? You can use the results for Geometric series with complex numbers 2? ?? 2? ? = ?? 2? ? ? 2? Replace denominator with 2???? ?? 2? ? 2? 2? = ? ?+ ???? ? ?, Given that ? = ??? where ? is a positive integer, show that: 2???? ? 2? 1 + ? + ?2+..+?? 1= 1 + ???? 1 2? ???? ? ???= ????? You will need to use the exponential form of ? in order to then use the patterns we just saw Multiply by 2i ???? ? ???= 2?????? Replace ?? with ? ???? + ????? = ??? 2? ? 2? ?? 2? ? ?? ? 2?= 2???? ? ? ? ? = ??? ??? + ???? ? ?? ? = ? (don t confuse ?symbols with the letter n !) 1E
? 1 ???= ? + ?? + ??2+..+??? 1=? 1 ?? ? ???= ? + ?? + ??2+..= 1 ?, ? < 1 ???= 1 1 ? ?=0 ?=0 1 2 1 2? Complex Numbers ????+ ? ???= ????? ???? ? ???= ????? ?? 2? You can use the results for Geometric series with complex numbers 2? ?? 2? ? = ?? 2? ? ? 2? Replace denominator with 2???? ?? 2? ? 2? 2? = ? ?+ ???? ? ?, Given that ? = ??? where ? is a positive integer, show that: 2???? Divide all by 2 ?? 2? ? 2? ? ? 2? = 1 + ? + ?2+..+?? 1= 1 + ???? ???? You can rewrite like this (you will not need to do this everytime, but it helps to show what happens next ) ?? 2? ? 2? You will need to use the exponential form of ? in order to then use the patterns we just saw =1 ? ? ??? ???? + ????? = ??? ? ? ? ? = ??? ??? + ???? ? Let 1 1 = ? ? ?= ? Let ? = ? ?? ? = ? 1 = ? ? So ? = ? Rewrite RHS (don t confuse ?symbols with the letter n !) 1 = ?2 So 1 This works! ?= ? 1 = 1 1E
? 1 ???= ? + ?? + ??2+..+??? 1=? 1 ?? ? ???= ? + ?? + ??2+..= 1 ?, ? < 1 ???= 1 1 ? ?=0 ?=0 1 2 1 2? Complex Numbers ????+ ? ???= ????? ???? ? ???= ????? ?? 2? You can use the results for Geometric series with complex numbers 2? ?? 2? ? = ?? 2? ? ? 2? Replace denominator with 2???? ?? 2? ? 2? 2? = ? ?+ ???? ? ?, Given that ? = ??? where ? is a positive integer, show that: 2???? Divide all by 2 ?? 2? ? 2? ? ? 2? = 1 + ? + ?2+..+?? 1= 1 + ???? ???? You can rewrite like this (you will not need to do this everytime, but it helps to show what happens next ) ?? 2? ? 2? You will need to use the exponential form of ? in order to then use the patterns we just saw =1 ? ? ??? Replace 1 ? with ? ???? + ????? = ??? ?? 2? ? 2? ? ? ? ? ? = ??? = ? ??? + ???? ? ??? ?? ? Simplify = ? ?? 2? ?? = (don t confuse ?symbols with the letter n !) ? 2? ??? 1E
? 1 ???= ? + ?? + ??2+..+??? 1=? 1 ?? ? ???= ? + ?? + ??2+..= 1 ?, ? < 1 ???= 1 1 ? ?=0 ?=0 1 2 1 2? Complex Numbers ????+ ? ???= ????? ???? ? ???= ????? ?? 2? ?? You can use the results for Geometric series with complex numbers = ? 2? Remember to check what you are aiming for We need to have no e terms, as well as a cot term. So we need to replace e with trigonometry Use Euler s relation with ? = ??? ? 2? + ???? ? ? ??? 2? ? 2? ? ?+ ???? ? ?, = Given that ? = ??? where ? is a positive integer, show that: ? 2? ??? Use cos ? = cos? and sin( ?) = sin? (this will make all the angles the same) ? 2? ???? ? 2? ? ??? ? 2? 1 + ? + ?2+..+?? 1= 1 + ???? = ? 2? ??? Expand the bracket on the numerator (this will give a ?2 which equals +1) You will need to use the exponential form of ? in order to then use the patterns we just saw ? 2?+ ??? ??? ? 2? ???? = ? 2? ???? + ????? = ??? Simplify ? 2? ? ? ? ? = ??? = ???? + 1 ??? + ???? ? ?? ? = ? (don t confuse ?symbols with the letter n !) 1E
? 1 ???= ? + ?? + ??2+..+??? 1=? 1 ?? ? ???= ? + ?? + ??2+..= 1 ?, ? < 1 ???= 1 1 ? ?=0 ?=0 1 2 1 2? Complex Numbers ????+ ? ???= ????? ???? ? ???= ????? You can use the results for Geometric series with complex numbers The series: ???+ ?2??+ ?3??+..+???? is geometric with first term ???, common ratio ??? and ? terms. You can also use patterns where you separate the real and imaginary parts of a geometric series ??=? 1 ?? 1 ? Replace ? = ???, ? = ??? and ? = ? ??=???1 ???? 1 ??? Remember this can be written in this way as well ??=??????? 1 ??? 1 1E
? 1 ???= ? + ?? + ??2+..+??? 1=? 1 ?? ? ???= ? + ?? + ??2+..= 1 ?, ? < 1 ???= 1 1 ? ?=0 ?=0 1 2 1 2? Complex Numbers ????+ ? ???= ????? ???? ? ???= ????? You can use the results for Geometric series with complex numbers ??=???1 ???? 1 ??? ??=??????? 1 ??? 1 You can also use patterns where you separate the real and imaginary parts of a geometric series ???+ ?2??+ ?3??+..+???? Replacing each term using Euler s relation = ???? + ????? + ???2? + ????2? + ???3? + ????3? +..+ ????? + ?????? Grouping real and imaginary parts separately = ???? + ???2? + ???3?+..+????? + ? ???? + ???2? + ???3?+..+????? Real Imaginary Therefore: The sum of the sequence on the left side is the real part of the formula on the right side ???1 ???? 1 ??? ???? + ???2? + ???3?+..+????? = ?? The sum of the sequence on the left side is the imaginary part of the formula on the right side ???1 ???? 1 ??? ???? + ???2? + ???3?+..+????? = ?? 1E
???1 ???? 1 ??? ???1 ???? 1 ??? ???? + ???2? + ???3?+..+????? = ?? ???? + ???2? + ???3?+..+????? = ?? Complex Numbers ??=???1 ???? 1 ??? ??=??????? 1 ??? 1 You can use the results for Geometric series with complex numbers So using this formula should lead to fewer steps! The restriction here is because if ? was a multiple of 2?, then the denominator of the result would be 0 ? = ???+ ?2??+ ?3??+..+?8??, for ? 2??, where ? is an integer 9?? 2 ???4? Show that ? =? a) ? 2 ??? The denominator contains sine, which means we are probably going to use the formula to the right to replace it as in the previous example 1 2? ???? ? ???= ????? For this you will need to use one of the formulae for the sum of a complex series shown above Either will work, but to make it easier, consider the denominator The power on the left needs to be higher than the power on the right, if it is written this way 1E
???1 ???? 1 ??? ???1 ???? 1 ??? ???? + ???2? + ???3?+..+????? = ?? ???? + ???2? + ???3?+..+????? = ?? Complex Numbers ??=???1 ???? 1 ??? ??=??????? 1 ??? 1 ??=??????? 1 ??? 1 You can use the results for Geometric series with complex numbers In our question, ? = 8 ???8 1 ??? ? = ???+ ?2??+ ?3??+..+?8??, for ? 2??, where ? is an integer = ??? 1 Rewrite =????8?? 1 ??? 1 Multiply the numerator and denominator by ? ?? on the denominator 9?? 2 ???4? Show that ? =? a) 2, in order to create the pattern we need ? 2 ?? 2 ?8?? 1 ??? =? ?? 2 ? ?? ? 2 Replace the denominator using the pattern to the left 1 2? ?? 2 ?8?? 1 2????? ???? ? ???= ????? =? Multiply by 2i To create a similar pattern on the numerator, factorise out ?4?? from the bracket 2 ???? ? ???= 2?????? Replace ? with 1 ?? 2?4???4?? ? 4?? 2????? 2 =? ? 2 ?? 2 ? ?? ? 2 = 2???? 2 1E
???1 ???? 1 ??? ???1 ???? 1 ??? ???? + ???2? + ???3?+..+????? = ?? ???? + ???2? + ???3?+..+????? = ?? Complex Numbers ??=???1 ???? 1 ??? ??=??????? 1 ??? 1 ?? 2?4???4?? ? 4?? 2????? You can use the results for Geometric series with complex numbers =? 2 Simplify/replace terms on the numerator 9?? 2 2???? 4? 2????? ? = ???+ ?2??+ ?3??+..+?8??, for ? 2??, where ? is an integer =? 2 Divide numerator and denominator by 2? 9?? 2 ???4? Show that ? =? a) 9?? 2 ??? 4? ???? =? ? 2 ??? 2 1 2? ???? ? ???= ????? Multiply by 2i ???? ? ???= 2?????? Replace ? with 4 ?4?? ? 4??= 2???? 4? 1E
???1 ???? 1 ??? ???1 ???? 1 ??? ???? + ???2? + ???3?+..+????? = ?? ???? + ???2? + ???3?+..+????? = ?? Complex Numbers ??=???1 ???? 1 ??? ??=??????? 1 ??? 1 9?? 2 ???4? ???? ? You can use the results for Geometric series with complex numbers ? = ?? 2 Replace the exponential term using Euler s relation 9? 2 + ????9? ? = ???+ ?2??+ ?3??+..+?8??, for ? 2??, where ? is an integer ??? ???4? 2 = ?? ???? 2 For ?, we are only considering the real part 9?? 2 ???4? Show that ? =? a) ???9? ? 2 ???4? ??? 2 = ???? From this, ? is the real part of the sum of ?, which we showed in part a) 2 Let: Rewrite ? = ???? + ???2? + ???3?+..+???8? ? = ???? + ???2? + ???3?+..+???8? 9? 2 ? 2 = ??? ???4?????? b) Use your answer to part a to show that ? = ???9? similar expressions for ? and ? 2???4??????? 2, and find ? 1E
???1 ???? 1 ??? ???1 ???? 1 ??? ???? + ???2? + ???3?+..+????? = ?? ???? + ???2? + ???3?+..+????? = ?? Complex Numbers ??=???1 ???? 1 ??? ??=??????? 1 ??? 1 9?? 2 ???4? ???? ? You can use the results for Geometric series with complex numbers ? = ?? 2 Replace the exponential term using Euler s relation 9? 2 + ????9? ? = ???+ ?2??+ ?3??+..+?8??, for ? 2??, where ? is an integer ??? ???4? 2 = ?? For ?, we are only considering the imaginary part (you don t need to include the ? itself since the expression for Q on the left does not) ???? 2 9?? 2 ???4? Show that ? =? a) ???9? ? 2 ???4? ??? 2 = ???? 2 Let: Rewrite From this, ? is the imaginary part of the sum of ?, which we showed in part a) 2 2 ? = ???? + ???2? + ???3?+..+???8? ? = ???? + ???2? + ???3?+..+???8? 9? ? = ??? ???4?????? b) Use your answer to part a to show that ? = ???9? similar expressions for ? and ? 2???4??????? 2, and find ? 1E
???1 ???? 1 ??? ???1 ???? 1 ??? ???? + ???2? + ???3?+..+????? = ?? ???? + ???2? + ???3?+..+????? = ?? Complex Numbers ??=???1 ???? 1 ??? ??=??????? 1 ??? 1 You can use the results for Geometric series with complex numbers 9? 2 ? 2 9? 2 ? 2 ? = ??? ???4?????? ? = ??? ???4?????? ? = ???+ ?2??+ ?3??+..+?8??, for ? 2??, where ? is an integer ???9? ? 2 ? 2 ???4?????? ? ?= 2 ???9? ???4?????? 2 9?? 2 ???4? Simplify Show that ? =? a) ? 2 ??? ? ?= ??? 9? 2 Let: ? = ???? + ???2? + ???3?+..+???8? ? = ???? + ???2? + ???3?+..+???8? b) Use your answer to part a to show that ? = ???9? similar expressions for ? and ? 2???4??????? 2, and find ? 1E
