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Sequences & Series

Σ summation notation

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IB Math AISL

Sequences & Series

Σ summation notation

Edit

Exercises

Key Skills

Σ summation notation - Exercises | IB Math AISL | Perplex

Σ summation notation

Summation notation, properties of sums

Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)

Exercises

No exercises available for this concept.

Practice exam-style σ summation notation problems

Key Skills

Understanding summation notation

SL Core 1.2

As a shortcut for writing out long sums, we can use the symbol ∑ with the following "settings":

n=(start value)∑end value of n(term depending on n)

Here n is called the index, but other letters can also be used in place of n.

Sum of a constant

SL Core 1.2

k=1∑nc=c+c+⋯+c=n⋅c🚫

Sums with scalar multiples

SL Core 1.2

k=1∑ncak=ck=1∑nak🚫

Sum of a sum

Essentially, any ∑ of a sum can be broken into two ∑ 's.

k=1∑n(ak+bk) =(a1+b1)+(a2+b2)+⋯+(an+bn)=a1+a2+⋯an+b1+b2+bn=k=1∑nak+k=1∑nbk

Or vice versa! If two sums have the same start and stop index (eg k=1 up to n), they can be merged.

Splitting a sum in Σ form

SL Core 1.2

For any series of the form k=1∑nak and any integer m between 1 and n, we can split the series at the index m:

k=1∑nak=k=1∑mak+k=m+1∑nak

Understanding summation notation

SL Core 1.2

As a shortcut for writing out long sums, we can use the symbol ∑ with the following "settings":

n=(start value)∑end value of n(term depending on n)

Here n is called the index, but other letters can also be used in place of n.

Sum of a constant

SL Core 1.2

k=1∑nc=c+c+⋯+c=n⋅c🚫

Sums with scalar multiples

SL Core 1.2

k=1∑ncak=ck=1∑nak🚫

Sum of a sum

Essentially, any ∑ of a sum can be broken into two ∑ 's.

k=1∑n(ak+bk) =(a1+b1)+(a2+b2)+⋯+(an+bn)=a1+a2+⋯an+b1+b2+bn=k=1∑nak+k=1∑nbk

Or vice versa! If two sums have the same start and stop index (eg k=1 up to n), they can be merged.

Splitting a sum in Σ form

SL Core 1.2

For any series of the form k=1∑nak and any integer m between 1 and n, we can split the series at the index m:

k=1∑nak=k=1∑mak+k=m+1∑nak