Why Coefficients Must Be Aligned
A power series is organized by powers of a centered variable. When two series have the same center, their terms can be added by matching the coefficients of the same power. This sounds simple, but a sparse series may omit many powers. If the omitted powers are not made explicit, it is easy to add terms according to their positions in a written list rather than according to their degrees.
For example, the first term of \(1+2x^2+3x^4+\cdots\) is a constant, while the first term of \(3x+4x^3+5x^5+\cdots\) has degree one. They are not like terms. To add the series correctly, first identify the coefficient at degree zero, degree one, degree two, and so on. A missing power has coefficient zero.
Coefficients Indexed by Degree
A power series centered at \(c\) has the form \(\sum_{n=0}^{\infty}a_n(x-c)^n\), where \(a_n\) is the coefficient of degree \(n\). The same series may be written in a sparse form that displays only the nonzero terms. The full coefficient form is useful for addition because it gives every degree its own position, including degrees whose coefficients are zero.
The zero coefficient does not add a term’s value to the series, because \(0(x-c)^n=0\). Its purpose is to preserve the degree-by-degree organization. Once each series is written in that form, addition is coefficientwise: the coefficient at degree \(n\) in the sum is the sum of the two coefficients at degree \(n\).
This equation applies at values of \(x\) where both series converge. The task in this tutorial is to determine the coefficients on the right accurately; questions about how far from the center the sum converges belong to a separate analysis.
Worked Example: Adding an Even-Power Series
Worked Example: Adding an Even-Power Series
Add \(A(x)=1+3x^2+5x^4+7x^6+\cdots\) and \(B(x)=2x+3x^2+4x^3+5x^4+\cdots\), and write the result by degree.
The series \(A\) has only even powers. Its coefficient at each odd degree is therefore zero. The displayed terms give the coefficient list by degree as \(1,0,3,0,5,0,7,\ldots\). For \(B\), the coefficient of \(x^0\) is zero, followed by coefficients \(2,3,4,5,6,\ldots\). Now add corresponding entries:
Thus the sum begins \(1+2x+6x^2+4x^3+10x^4+6x^5+14x^6+\cdots\). The coefficients at degrees two and four include contributions from both series; at odd degrees, \(A\)'s contribution is zero.
Check: Expanding the original terms through degree six gives \((1+3x^2+5x^4+7x^6)+(2x+3x^2+4x^3+5x^4+6x^5+7x^6)\). Combining equal powers gives \(1+2x+6x^2+4x^3+10x^4+6x^5+14x^6\), which matches the aligned coefficients.
A Systematic Alignment Method
When a series is displayed sparsely, make a coefficient list indexed by the actual powers rather than by the order in which the terms appear. If a term is missing at degree \(n\), record zero in that degree’s position. Then add entries with the same index. A table is often the clearest approach when the series have several missing degrees.
Write each term as a coefficient times \((x-c)^n\), and note its degree \(n\).
Make a row of coefficients for each series, with one entry for each degree in the range being added.
For every \(n\), calculate \(a_n+b_n\), not the sum of terms that happen to be next to one another.
Restore the powers and compare several initial terms with the original sum.
Worked Example: Missing Degrees in Both Series
Worked Example: Missing Degrees in Both Series
Add the finite power series \(P(x)=2-x+4x^3-2x^6\) and \(Q(x)=5x^2+3x^3+x^5\).
List degrees zero through six. The coefficients of \(P\) are \(2,-1,0,4,0,0,-2\); its degrees two, four, and five are absent. The coefficients of \(Q\) are \(0,0,5,3,0,1,0\); its degrees zero, one, four, and six are absent. Add by column:
The degree-four coefficient is zero in both series, so the sum has no degree-four term either. The result is \(2-x+5x^2+7x^3+x^5-2x^6\).
Check: Adding the original expressions gives \(2-x+4x^3-2x^6+5x^2+3x^3+x^5\). Collecting by degree gives constant \(2\), degree one \(-1\), degree two \(5\), degree three \(4+3=7\), degree four \(0\), degree five \(1\), and degree six \(-2\). This reproduces the stated result.
Shifted Summation Indices
A common source of misalignment is a summation index that does not start at zero. The index in a summation is a label; it is not automatically the degree of the power. For example, in \(\sum_{k=0}^{\infty}d_k(x-c)^{k+1}\), the index \(k\) starts at zero, but the first power has degree one. To align this series with another series indexed by degree \(n\), identify which \(k\) produces degree \(n\).
More generally, if a term contains \((x-c)^{k+r}\), then its degree is \(n=k+r\). Therefore \(k=n-r\), and its contribution at degree \(n\) is the coefficient originally labeled \(d_{n-r}\), whenever that index is in the series’ allowed range. At degrees below the first represented power, the contribution is zero.
Worked Example: Translating Shifted Indices
Worked Example: Translating Shifted Indices
Add \(A(x)=\sum_{n=2}^{\infty}(n-1)x^n\) and \(B(x)=\sum_{k=0}^{\infty}2(k+1)x^{k+1}\), writing the answer with coefficients indexed by the power of \(x\).
In \(A\), the index \(n\) is already the power. There are no terms of degrees zero and one, so the coefficients there are zero. For \(B\), the power is \(k+1\). Set \(m=k+1\), so its terms have degree \(m\) and coefficient \(2m\); the index condition \(k\geq0\) means \(m\geq1\). Thus \(B\)'s degree-zero coefficient is zero and its coefficient at degree \(n\geq1\) is \(2n\).
At degree zero, the sum coefficient is \(0+0=0\). At degree one, it is \(0+2=2\). For every degree \(n\geq2\), the contributions are \(n-1\) from \(A\) and \(2n\) from \(B\), giving \((n-1)+2n=3n-1\). Therefore the sum can be written as
Check: The first terms of \(A\) are \(x^2+2x^3+3x^4+4x^5+\cdots\). The first terms of \(B\), using \(k=0,1,2,3,\ldots\), are \(2x+4x^2+6x^3+8x^4+10x^5+\cdots\). Their sum begins \(2x+5x^2+8x^3+11x^4+14x^5+\cdots\). The formula gives \(3(2)-1=5\), \(3(3)-1=8\), \(3(4)-1=11\), and \(3(5)-1=14\), as required.
Why Coefficient Alignment Works
The justification is that a finite partial sum is an ordinary sum of terms, and ordinary addition groups terms with the same power. For a power series, take the partial sums through degree \(N\). At each degree from zero to \(N\), the coefficient in their sum is exactly the sum of the two coefficients at that degree. If both series converge at \(x\), their partial sums converge to their respective sums, and the sums of the partial sums converge to the sum of those limits. This gives the coefficientwise addition rule wherever both original series converge.
For sparse series, coefficients omitted from the written expression are inserted as zero before this calculation. The identity then works without a special case for even powers, gaps, or differing starting indices. The only essential bookkeeping is to associate each coefficient with the correct degree.
A Common Mistake
A frequent error is to add the first displayed terms together, then the second displayed terms together, without checking whether their powers match. For instance, one might incorrectly write
This adds coefficients by position in the displayed lists: \(1+3\), \(2+4\), and \(3+5\). But the terms in each of those pairs have different degrees. The constant \(1\) cannot be combined with \(3x\), and \(2x^2\) cannot be combined with \(4x^3\). The correct alignment is
The correction is to insert zero coefficients at missing degrees and add only coefficients attached to the same power. This method applies to power series with a common center. If the centers differ, the expressions are not in a directly aligned form; do not apply this coefficient table as though their powers were powers of the same centered variable.
Check Your Understanding
For each question, identify coefficients by degree rather than by the order in which terms are displayed.
- Write the coefficient list for \(4+2x^2-x^6\) from degree zero through degree six.
- What is the coefficient at degree five in \(1+3x^2+7x^4+9x^6+\cdots\)?
- In \(\sum_{k=1}^{\infty}k x^{k+2}\), what is the first degree that occurs, and what is its coefficient?
- Add \(2+x^3\) and \(5x+4x^3\), showing the coefficients from degree zero through degree three.
- Explain why terms that are adjacent in two written sparse series need not be like terms.
