Question

proceed as in example 3 in section 6.1 to rewrite the given expression using a single power series whose general term involves xk. [infinity] 4ncnxn − 1 n = 1 [infinity] 7cnxn 1 n = 0

Answer 1

To unify the terms, we rewrite the expression as a single series, combining similar terms and ensuring the general term involves xⁿ. The final result is Σ(4cₙₓⁿ - 7cₙₓⁿ⁻¹) from n=1 to ∞. This simplification enhances the expression's clarity and facilitates further mathematical analysis.

Step-by-step explanation:

To rewrite the expression using a single power series involving xᵏ, we can combine the two series into one. The given expression is Σ(4cₙₓⁿ - 7cₙₓⁿ⁻¹) from n=1 to ∞. Notice that the term 7cₙₓⁿ⁻¹ is similar to the term 4cₙₓⁿ but with xⁿ replaced by xⁿ⁻¹. To unify these terms, we can rewrite the expression as Σ(4cₙₓⁿ - 7cₙₓⁿ⁻¹) from n=1 to ∞.

In the new expression, the general term involves xⁿ, which meets the requirement of expressing the given series as a single power series in terms of xᵏ. By rearranging and combining terms, we achieve a simplified expression that captures the essence of the original series. This technique of combining terms is a common approach in manipulating power series to express them in a more compact and convenient form.

In conclusion, the given expression can be succinctly expressed as Σ(4cₙₓⁿ - 7cₙₓⁿ⁻¹) from n=1 to ∞, where the general term involves xⁿ. This provides a single power series representation for the given series, making it more amenable to analysis and further mathematical operations.

Answer 2

To rewrite the provided expressions into a single power series, one must identify a common general term, possibly by adjusting indexes and utilizing binomial theorem principles. Algebraic manipulation will be used to combine series with terms involving different powers of x.

Step-by-step explanation:

The student's question seems to pertain to rewriting a given expression to involve a single power series. The provided expressions resemble separate series where one includes coefficients with xn, while the other with coefficients with
`x^(n-1)`. In mathematics, particularly in calculus or advanced algebra, combining series into a single series involves finding a common general term, which often requires utilizing properties of binomial theorem or algebraic manipulation.

To rewrite the given expression using a single power series, we would identify the pattern of each term and adjust the indexes if necessary so that both series have the same powers of x across their terms. Leveraging the binomial theorem and algebraic properties such as factorization or distributive law can be essential in combining the series.

For example, if we have a series
`\sum c_nx_n` starting at n=0 and another series
`\sum 4n_(c_n)x^(n-1)`starting at n=1, we might adjust the second series by changing the index to n=k+1, which would then allow us to match the powers of x across both series. After finding a common expression for the general term involving xk, we can sum both series into one single series representation.