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The phrase "a series of exchanges" is correct and usable in written English.
It can be used to describe multiple interactions or transactions between parties, often in a context of communication or trade.
Example: "The negotiation involved a series of exchanges that ultimately led to a mutually beneficial agreement."
Alternatives: "a sequence of interactions" or "a chain of transactions."
Exact(51)
After a series of exchanges, the committee ended its review, apparently satisfied.
Mr. Almunia also said that he expected to hold a series of "exchanges" with the search giant.
Shankland forced a series of exchanges that led to an easily winning endgame, so Meier gave up.
(You can find a series of exchanges that stakes out the positions of each side in great detail here).
Later, Stillman travelled frequently to China as part of a series of exchanges between American and Chinese weapons scientists.
All this led to a series of exchanges that grew in intensity as the debate rolled on.
Similar(9)
Citi recently completed a series of exchange offers that gave the company unquestioned financial strength.
From 1955 onward, however the British economy was bedevilled by a series of exchange crises which seemed to show that sterling could only be defended in a period of fixed exchange rates through strict control of the money supply.
Limited consular activities resumed in February, and there has been a series of exchange visits, including a trip to Tehran by a British parliamentary delegation led by former foreign secretary Jack Straw.
In late June, Barclays rolled out a series of exchange-traded notes, or E.T.N.'s, intended to reflect prices on a handful of individual commodities, including coffee, cocoa and sugar.
The key element in these changes was not money, but a series of exchange visits between the two communities that introduced Dubna's political leaders and medical professionals to more up-to-date methods and techniques.
More suggestions(17)
a series of discussions
a host of exchanges
a series of communications
a series of challenges
a series of trades
a series of interactions
a series of interfaces
a series of share
a series of discussion
a series of trading
a series of exchanged
a series of interaction
a series of market
a series of shared
a succession of exchanges
a number of exchanges
a barrage of exchanges
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com